Cécile L'héritier

Abdelhak Imoussaten

Sébastien Harispe

Gilles Dusserre

Benoît Roig

Identifying Criteria Most Influencing Strategy Performance: Application to Humanitarian Logistical Strategy Planning

Keywords: Multi-Criteria Decision Analysis (MCDA), Lessons Learned, Electre tri, Influencing Criteria Identification, Humanitarian domain

A growing interest is expressed by organizations for the development of approaches enabling to take advantage of past experiences to improve their decision processes; they may be referred to as Lessons Learned (LL) processes. Within the LL processes implementation framework, the development of semi-automatic approaches able to distinguish criteria having major influence on the evaluation of experiences is crucial for identifying relevant recommendations and performing efficient prescriptive analysis. In this paper, we propose to contribute to LL study by focusing on the definition of an approach enabling, in a specific setting, to identify the criteria most influencing the decision process regarding the overall performance evaluation of a reduced set of experiences. The proposed approach is framed on Multi-Criteria Decision Analysis, and specifically is based on the Electre tri method. In this paper, an illustration of the proposed approach is provided studying the evaluation of logistical response strategies in humanitarian emergency situations.

Introduction

Lessons Learned process (LL) 1 refers to a general Knowledge Management approach aiming at ensuring and improving proper functioning of organizations by collecting, analysing, disseminating and reusing tacit experiential knowledge. One of the main focuses of LL is therefore to study how to improve techniques enabling to discover and further take advantage of expert knowledge that is not explicitly formalized into organizations' Information Systems. In this context, LL is particularly interested in taking advantage of knowledge that is most often difficult if not impossible to formalize nowadays -in particular because of the technical difficulties to systematically and efficiently formalize such knowledge considering every-day practices used in organizations, e.g. domain experts often take decisions based on complex bodies of professional histories. Importance of LL is thus critical to fight against sensitive losses of expertise in organizations, and to achieve well-supported and understood decision-making processes. LL can indeed help improving both the performance and the quality of organizations by analysing past experiences and by using problem-solving methods -narrow links with well-established related domains such as Case Based Reasoning, Multiple-Criteria Decision Analysis and Knowledge Management/Representation exist; some of them will further be discussed.

In this paper, we contribute to LL study by focusing on the definition of an approach enabling, in a specific setting, to identify the criteria most influencing the decision process regarding the overall performance evaluation of a reduced set of experiences. Identifying the criteria most impacting performance is indeed of major importance for numerous applications related to LL; it can also be central for the global implementation of the LL process, e.g. it indeed helps a lot to identify relevant recommendations and to perform efficient prescriptive analyses. Identifying such criteria in real-world problems can however be challenging. We indeed here consider -as it is most often the case in numerous complex decision processes -that no explicit and formal definition of performance based on criteria analysis can be provided by domain experts. Experts can indeed evaluate performance but are not able to explicitly define the way it is evaluated, i.e. to define the model on which the evaluation is based, if any. We also consider the following common constraints: only a reduced set of observations is available 2and only limited interactions with domain experts can be expected. Additionally, we consider that it is most often unthinkable to perform automatic deep analyses of all pieces of information that could be related to each criterion to further distinguish relevant teachings. Therefore, based on the analysis of a reduced set of evaluated past experiences characterized by their performance on criteria, we propose a general approach framed on Multiple-Criteria Decision Analysis (MCDA) in order to distinguish the minimal set of criteria most impacting performance. The proposed approach is built upon Electre tri, a MCDA method enabling classifying a set of alternatives into predefined categories.

An application of the proposed approach to the humanitarian field is presented; it is used for analysing logistical response strategies in emergency situations. Our case study rests upon a corpus of past missions carried out by a Non-Governmental Organization (NGO); the core mission of the NGO is to deliver emergency aid to people affected by disasters (earthquake, tsunami, conflict, famine, epidemic. . . ). Our general goal is therefore to exploit knowledge that can be extracted from prior missions in order to formulate relevant recommendations that could be used for better defining future logistical response strategies. To this aim, this paper focuses on identifying the set of criteria most impacting performance evaluation of logistical response strategies. The paper is structured as follows: Section 2 briefly introduces state of the art related to LL and motivates the use of MCDA for extracting knowledge from past experiences. Section 3 presents our contribution for identifying the criteria most impacting performance evaluation; an introduction to Electre tri, the MCDA method used in our work, is also given in that section. Section 4 illustrates the use of the proposed approach for studying the logistical response in humanitarian emergency situations. Section 5 concludes this paper and highlights future work.

Related Works and Positioning

Whatever the field of application, learning from experience is a growing concern for numerous organizations. In this context, several approaches have emerged to fulfil the different LL needs expressed by organizations; they differ both in (i) the nature and the heterogeneity of information to collect, as well as in (ii) the expected purpose: exploitation, reuse, capitalization, and dissemination of knowledge. Various types of LL approaches can therefore be identified in the literature. Some of them rely on databases composed of past experiences and take advantage of domain-specific querying strategies, e.g. REX method [START_REF] Malvache | Mastering corporate experience with the rex method[END_REF]. Other approaches take advantage of knowledge models -using techniques related to Knowledge Management and Knowledge Representation (KM/KR) such as CommonKads [START_REF] De How | The commonkads organization model: content, usage and computer support[END_REF], or MKSM (Methodology for Knowledge System Management) [START_REF] Ermine | Méthode pour la gestion des connaissances. Ingénierie des systèmes d?information[END_REF]. Another category of LL approaches also considers the use of problem-solving methods. These methods, in particular Case-Based Reasoning (CBR) [START_REF] Aamodt | Case-based reasoning: Foundational issues, methodological variations, and system approaches[END_REF], rely on the use of knowledge extracted from previously experienced cases to solve similar new cases. A significant number of cases is however generally required to distinguish analogies using these methods.

In our study, we are considering LL towards an application to the logistical emergency response carried out by humanitarian organizations. In this domain, data collection is even more difficult, e.g. oral testimony often prevails, analyses most often rest upon reduced corpora of past missions. With this in mind, another challenge arises: to exploit and reason on a small number of complex past experiences. These experiences are composed of successions of decisions being influenced by different factors such as: the intuition, the context-dependent nature of the missions -instable environment, unpredictable events affecting the system -the multiple, and sometimes conflicting objectives. Nevertheless, despite decision-making complexity, according to the collaborating NGO, the logistical strategy of several missions could be, to a certain extent, generalized and reproduced. In that context, the NGO has established a procedure to learn lessons from past experiences, particularly from choices made in missions. It aims at providing an a posteriori evaluation of singular cases entailing a noteworthy success or a significant failure. To do so, a board of multidisciplinary experts conducts an analysis of the case based on related data and complementary interviews; then a conclusive report is provided. However, according to the NGO stakeholders, provided outcomes often face major weaknesses -e.g. hardly exploitable, lack of sharpness -the time-consuming nature of the process is also pointed out.

An open challenge is thus offered to LL automation for knowledge extraction from past experiences in NGOs. Recall however that in the humanitarian domain, relevant and accurate data is difficult to obtain, and cannot easily be automatically analysed -data access and analysis often require costly staff and experts involvement. A further constraint is therefore, while maintaining interactions with domain experts and decision-makers (DMs), to minimize information requests in order to lessen their involvement (both in terms of time and cognitive effort). All these constraints underline the need for a prior identification of relevant criteria in order to reduce the scope of analysis for applying deeper data analyses. In practice, DMs always -at least implicitly -consider a set of objectives that the mission has to reach in order to meet success. In addition, a set of relevant criteria can be considered to measure, for a specific mission, the degree of success of each objective, i.e., performance. Such a performance evaluation can however be a complex process. Indeed, to be successful, for DMs (i) a mission does not necessarily require maximal performance with respect to each criterion, and (ii) the criteria may not have the same importance. We therefore propose to take advantage of MCDA techniques to identify the subset of criteria most impacting the overall performance assessment of a specific mission.

The best known MCDA approaches are those derived from Multi-Attribute Value/Utility Theory (MAVT/MAUT) [START_REF] Dyer | Multiattribute utility theory (maut)[END_REF] and outranking methods [START_REF] Figueira | Electre methods[END_REF]. Since every MCDA method differently combines performance and criteria importance to define the overall performance of a mission, the identification of the subset of criteria of interest will necessarily depend on the selected MCDA method. In this work, we have chosen to use Electre tri, a method from the well-known ELECTRE family methods [START_REF] Figueira | An overview of electre methods and their recent extensions[END_REF]. It has been selected according to three main criteria related to NGOs context: it is simple to reuse; non-compensatory, and techniques exist to reduce the amount of information asked to DMs while determining model parameters -agreement among DMs is considered here. The non-compensatory condition means that an excellent performance on specific criteria (e.g. cost) cannot compensate for a bad performance on other prevailing criteria (e.g. degree of achievement of the medical objectives). In order to reduce the amount of information that has to be provided, Electre tri relies on indirect identification of the parameters. In addition, Electre tri is an outranking method: instead of scoring alternatives, outranking methods are based on pair comparisons. Indeed, it is more natural for DMs to compare the strength and the weakness of two alternatives rather than assigning an arbitrary value to each alternative, and comparing them subsequently. Interestingly, outranking methods are based on relatively weak mathematical assumptions [START_REF] Vincke | Multicriteria decision-aid[END_REF]. Nevertheless, obtaining the model parameters is a very difficult issue because of the non-linearity due to the thresholds induced in the model. However some recent works have proposed simplified assumptions and a procedure to determine the parameters of the Electre tri method has been proposed [START_REF] Mousseau | Inferring an electre tri model from assignment examples[END_REF].

Determining Criteria Contribution within Electre tri

The following section first introduces the Electre tri method and is then dedicated to the presentation of our approach: the identification of the subset criteria having the major contribution on the overall performance of a mission.

General approach

The approach proposed in this paper aims to determine the criteria on which LL process should focus when a mission is analysed. To facilitate DMs' involvement, the information requirement is minimized and the cognitive effort reduced as much as possible. For instance, the parameters of the MCDA model used for the analysis will be transparent for DMs. They will only be asked to assign examples of fictive missions to the predefined categories. These examples of missions are defined by their performance on each evaluation criterion, such that DMs are able to easily classify them. The general approach can be decomposed as follows:

1. Determine the set of criteria on which missions will be evaluated; 2. Automatic identification of Electre tri parameters using a training set of fictive examples of missions; 3. Classify/Assign the mission to one of the predefined categories using this Electre tri model; 4. Determine the criteria strongly contributing to that classification.

Classifying alternatives using Electre tri

We introduce the technical details related to Electre Tri, the specific MCDA method used in our work.

Let A = {a 1 , . . . , a m } a set of alternatives, and F a set of criteria satisfying consistency conditions, i.e. completeness (all relevant criteria are considered), monotonicity (the better the evaluation of an alternative on considered criteria, the more it is preferable to another), and non-redundancy (no superfluous criteria are considered) [START_REF] Figueira | Electre methods[END_REF]. According to traditional MCDA outranking methods notations, we denote g j (a) the score of alternative a ∈ A on the criterion j ∈ F. For some pairs of alternatives a, b ∈ A, Electre tri builds an outranking relation aSb meaning that 'a is at least as good as b'. To this aim, the method uses concordance, comprehensive concordance, discordance and credibility indices [START_REF] Figueira | An overview of electre methods and their recent extensions[END_REF]. The concordance index is defined using the indifference q j (g j (a)) and preference p j (g j (a)) thresholds; these thresholds allow to take the imprecision of the evaluations g j (a) into account. 3Let's define three subsets

F c 1 (a, b), F c 2 (a, b) and F c 3 (a, b
) that divide criteria of F with regard to the comparison of a and b performances:

-F c 1 (a, b) = {j ∈ F : g j (a) ≤ g j (b) -p j (g j (b))}, -F c 2 (a, b) = {j ∈ F : g j (b) -p j (g j (b)) < g j (a) < g j (b) -q j (g j (b))}, -F c 3 (a, b) = {j ∈ F : g j (b) -q j (g j (b)) ≤ g j (a)}.
The partial concordance index on criterion j is defined using the previous subsets as follows: The comprehensive concordance index is then defined using the relative importance w j of each criterion j ∈ F:

∀a, b ∈ A, c j (a, b) =    0 if j ∈ F c 1 (a, b), α j (a, b) if j ∈ F c 2 (a, b), 1 if j ∈ F c 3 (a, b). (1
∀a, b ∈ A, c(a, b) = j∈F w j • c j (a, b) = j∈F c 3 (a,b) w j + j∈F c 2 (a,b) w j • α j (a, b) (2)
The discordance index is defined by introducing a veto threshold v j (g j (b)):

∀a, b ∈ A, d j (a, b) =    1 if j ∈ F d 1 (a, b), gj (a)-gj (b)+pj (gj (b)) pj (gj (b))-vj (gj (b)) if j ∈ F d 2 (a, b), 0 if j ∈ F d 3 (a, b). (3) where:
-

F d 1 (a, b) = {j ∈ F : g j (a) ≤ g j (b) -v j (g j (b))}, -F d 2 (a, b) = {j ∈ F : g j (b) -v j (g j (b)) < g j (a) < g j (b) -p j (g j (b))}, -F d 3 (a, b) = {j ∈ F : g j (b) -p j (g j (b)) ≤ g j (a)}.
The credibility index is further defined as follows:

ρ S (a, b) = c(a, b) • j∈F d (a,b) 1 -d j (a, b) 1 -c(a, b) (4)
where

F d (a, b) = {j ∈ F : d j (a, b) > c(a, b)}. Finally, the outranking relation aSb is considered when ρ S (a, b) ≥ λ, λ ∈ [0.5, 1] [9].
In Electre tri, the credibility index is used to assign alternatives to predefined categories. Suppose that alternatives should be assigned to p categories {C 1 , ...C p } defined by p -1 profiles {b 1 , ...b p-1 }. Each profile b h , h ∈ {1, . . . , p -1}, is the upper limit of C h and the lower limit of C h+1 . Let a be an alternative to assign to a category C h , h ∈ {1, . . . , p}. Two assignment procedures are possible:

-Pessimistic procedure: a is compared first to profiles defining best categories until meeting the first b h such that aSb h , then a is assigned to C h+1 . -Optimistic procedure: a is compared first to profiles defining worst categories until meeting the first b h such that b h Sa, then a is assigned to C h .

Note that Electre tri requires determining the following parameters:

-the profiles of the categories, defined by their evaluations g j (b h ), ∀j ∈ F, ∀h ∈ {1, . . . , p}, -the weight coefficients, w j , ∀j ∈ F, -indifference, preference and veto thresholds, q j (g j (b h )),p j (g j (b h )), v j (g j (b h )), ∀j ∈ F, ∀h ∈ {1, . . . , p}, -the cutting level λ.

These parameters define the preference model of the DMs. The direct elicitation of these parameters values, i.e. asking directly the DMs to provide them, remains difficult since they do not correspond to the usual language or indicators that DMs use to express their opinion and expertise. To overcome this issue, procedures have been proposed to infer the parameters of Electre tri model. Indeed, in [START_REF] Mousseau | Inferring an electre tri model from assignment examples[END_REF] this inference is based on the assignment of a set of examples of alternatives A * for which DMs have a clear preference. This is done by resolving a non-linear programming problem, given in equation ( 5), where Electre tri parameters are the variables. The slack variables x k and y k are introduced to transform the inequality constraints ρ(a k , b h k -1 ) ≥ λ and ρ(a k , b h k ) ≤ λ into equality constraints. Further details about variables and constants involved in equation ( 5) are given in [START_REF] Mousseau | Inferring an electre tri model from assignment examples[END_REF].

α + a k ∈A * (x k + y k ) → max, such that: (5) α ≤ x k and α ≤ y k , ∀a k ∈ A * m j=1 w j c j (a k , b h k -1 ) m j=1 w j -x k = λ, ∀a k ∈ A * m j=1 w j c j (a k , b h k ) m j=1 w j + y k = λ, ∀a k ∈ A * λ ∈ [0.5, 1] g j (b h+1 ) ≥ g j (b h ) + p j (b h ) + p j (b h+1 ), ∀j ∈ F, ∀h ∈ B p j (b j ) ≥ q j (b h ), ∀j ∈ F, ∀h ∈ B w j ≥ 0, w j ≤ 1 2 m j=1
w j , q j (b h ) ≥ 0, ∀j ∈ F, ∀h ∈ B

Determining Criteria of interest

Let consider a the alternative defined by the performance of the currently evaluated mission on all evaluation criteria. The alternative a is classified as "good", "medium" or "bad" by DMs. The aim of this section is to propose a method to determine which subset of criteria strongly contributes to the overall result of a. In our setting, this alternative is assigned to one of the categories using Electre tri; this problem is therefore equivalent to identifying the subset of criteria strongly contributing to the credibility index of a when it is compared to the profiles determining the category finally assigned to a.

As explained above, when an outranking method is used, the contribution of criteria to the overall score is the portion of their weights considered in the comprehensive concordance index. This contribution depends on the belonging of criteria to one of the three subsets F c 1 , F c 2 and F c 3 . Naturally, when comparing a to a profile b, we define the contribution of a subset of criteria I ⊆ F to the comprehensive concordance index, as the quantity c I (a, b):

c I (a, b) = j∈I w j • c j (a, b) = j∈F c 3 (a,b)∩I w j + j∈F c 2 (a,b)∩I w j • α j (a, b) (6)
By convention c F (a, b) is c(a, b).
Let define two indices r, s ∈ {1, ..., p} obtained by the two procedures of Electre tri classification:

1. Pessimistic procedure: ∀h ∈ {r, ..., p}, not(aSb h ) and aSb r : a is classified in category C r ; 2. Optimistic procedure: ∀h ∈ {1, ..., s -1}, aSb h and b s Sa and not(aSb s ): a is classified in category C s .

Two cases have to be distinguished according to the quality of C s :

1. The category C s is considered as a "good" category. Then we have to determine criteria strongly contributing to the strength of a; 2. The category C s is considered as a "bad" category. Then we have to determine criteria strongly contributing to the weakness of a.

Knowing that the criteria contributing strongly to the weakness of a are those strongly contributing to the strength of b s , we only present the first case. When a is assigned to a "good" category, we have aSb r . Then for the cutting level λ ∈ [0.5, 1] defined by the DMs, ρ S (a, b r ) > λ means:

c(a, b r ) • j∈F d (a,br) 1 -d j (a, b r ) 1 -c(a, b r ) > λ (7) Let denote λ(a, b) = λ j∈F d (a,b) 1-d j (a,b) 1-c(a,b)
. The most contributing subset of criteria This section illustrates the use of the proposed approach in a practical case aiming at studying the logistical response in humanitarian emergency situations. We consider the context of LL in a humanitarian organisation case (see Section 2), it aims at determining, a posteriori, if choices were good or bad w.r.t objectives of an analysed mission, and to explain why. We deal with this problem by (i) analysing a real mission using Electre tri method which assigns the mission to a predefined category characterizing its overall success, (ii) identifying the criteria contributing to the mission success and those most impacting its performance assessment. In this case study, among the diversity of missions carried out by the collaborating NGO, we are focusing on distribution missions -distribution of medicines, food and shelters -which clearly entail similar logistics. For this illustrative example, we consider a mission a, corresponding to a mission largely inspired -but slightly simplified for the example -from a real food distribution mission carried out by the NGO. We define three categories:

I ⊆ F
-C 1 includes failed missions for which objectives have not been reached, -C 2 covers missions with met objectives but moderate success, -C 3 covers successful missions: those exceeding the objectives.

The performance of a mission is evaluated according to a set of eight criteria, i.e. F = {1, 2, ..., 8}, listed in Table 1. Criterion g 1 assesses deadline compliance, criterion g 3 assesses the portion of the logistical costs w.r.t the total costs and criterion g 4 the number of enlisted human resources. The preference decreases on these criteria. Criterion g 2 assesses the percentage of achievement of the targeted population: preference increase on it. Criteria g 5 , g 6 , g 7 and g 8 are assessed in a qualitative way using an ordinal scale with four levels: the scale is {Small, M edium, High, V ery High} for g 5 and g 6 , and {Bad, M oderate, Good, V ery Good} for g 7 and g 8 . We use the scale {1, 2, 3, 4} to encode these two scales. Preferences on the criteria will further be characterized by the definition of pseudo-criteria using thresholds q j , p j and v j .

Determination of Parameters

As a starting point, several parameters have to be fixed to determine the Electre tri model that best fits DMs' preferences. As mentioned in Section 3.2, an approach can be used to determine those parameters by solving a non-linear optimisation problem [START_REF] Mousseau | Inferring an electre tri model from assignment examples[END_REF]. Similarly to the work proposed in that paper [START_REF] Mousseau | Inferring an electre tri model from assignment examples[END_REF], to facilitate the optimisation problem setting and resolution, veto thresholds are not inferred, we assume that they are directly given by DMs (see Table 3). Therefore, the variables are: g j (b h ), w j , q j (g j (b h )), p j (g j (b h )), ∀j ∈ {1, . . . , 8}, ∀h ∈ {1, 2}, λ. The problem constraints are defined from parameters definition and the assignment of a training set:

-p j (g j (b h )) ≥ q j (g j (b h )) ≥ 0, λ ∈ [0.5, 1], -a k assigned to C h means that a k Sb h-1 and not(a k Sb h ).
Regarding the profiles of the categories, a training set of ten alternatives has been used (see Table 2); each alternative a k is assigned to one of the three categories by the DMs. These fictive missions a k , used as a training set to infer model parameters are distinct from the alternative a, a real mission that we aim at analysing here. Then in this context, we have to consider 58 variables and 90 constraints. Table 3 gives the values of parameters obtained from the resolution of the non-linear optimisation problem resulting from the training set analysis.

Criteria contribution

Let denote the analysed mission by a. DMs assess the performance of a on the criteria of F as shown in Table 4. We apply the pessimistic procedure of Electre tri using the identified model parameters (Table 3), and we consider the credibility indices ρ(a, b 1 ) = 0.995 and ρ(a, b 2 ) = 0.2. Then the mission a is assigned to category C 2 , since not(aSb 2 ), and aSb 1 with b 1 the lower profile of C 2 . Knowing that aSb 1 , we are looking for the subset of criteria I ⊆ F that most contributes to the establishment of this outranking relation, both because of the importance of the criteria, and the performance of a on these criteria. Table 4 shows the individual contribution c {j} (a, b 1 ) of each criterion g j to the global performance of a when it is compared to b 1 . Then in our Electre tri model, the cutting level is λ = 0.93 and the threshold γ = c(a,b1) 2 n = 0.004 is considered, n = |F|. Thus, solving the equation proposed in [START_REF] Vincke | Multicriteria decision-aid[END_REF], the subset of criteria I * = {2, 3, 4} is identified, with a maximal β * = 0.914. This identified subset has the contribution c I (a, b 1 ) = 0.85, stating that I * contributes to 91.4% of the global performance of a when it is compared to the reference profile b 1 . In the example, the subset of identified criteria I * corresponds to the subset of all criteria having a large weight (w 2 = 37.5, w 3 = 14, w4 = 18.3) w.r.t the others criteria having weights lower than 4. Thus, a outranks 1 on all of the criteria of I * . Conversely, the performance of a compared to b 2 on the important criteria gives I * ⊂ F c 2 (a, b 2 ) with α 2 (a, b 2 ) = 0.06, α 3 (a, b 2 ) = 6.10 -6 and α 4 (a, b 2 ) = 0.99: the bad performance of a compared to b 2 is then explained.

In the presented solution, the selection of criteria in I * is not affected by the defined threshold γ. Indeed, the coalition of criteria {2, 3, 4} having significant contributions is enough to exceed λ. In order to highlight the role of γ, lets consider β = 0.92, such that c I (a, b 1 ) has to exceed 0.86. It means that at least one criterion with a weak contribution (≥ 0.01) has to be selected in I * , e.g. = 0.004 is considered, it does not prevent the selection of any criterion with an insignificant contribution. Conversely, a large γ large = c(a,b1) n = 0.124, refuses the selection of criteria {1, 5, 6, 7, 8}, then the subset I * cannot be built. The higher the value of γ is, the more restrictive it is on the final identified subset of criteria.

It is important to mention here that we have applied the approach according to one selected combination of values for the parameters -local solution. Indeed, the programming problem solved to determine the parameters of the model has infinity of solutions, and thus the determined subset I * could change. This solution also depends on the initial values provided to the program. In addition, for illustration purpose a small training set has been built. Consequently, a throughout robustness analysis should be carried out in our future works.

This application aimed at analysing, a posteriori, the performance of a food distribution mission. The proposed approach allowed to identify three criteria: "Percentage of population reached w.r.t the targeted one", "Number of human resources enlisted", "Invested logistical costs". It means that, in similar contexts, a good performance on these three criteria could guaranty a successful mission. It is then relevant to focus on these elements for learning lessons. Furthermore, identifying this subset of criteria is of major interest to restrain the search space to perform deeper analyses. Indeed, in a further step our work aims at identifying, between similar distribution missions, the shared features explaining the success or failure of the mission to then infer general lessons. Thus, knowing the criteria mainly responsible for the achievement of a good or bad overall performance, allows to focus searching on properties influencing these outlined criteria. In more comprehensive approaches that we are targeting, dealing with up to twenty criteria, the use of such a procedure will be required.

Conclusion

Organizations today express a growing interest for semi-automatic approaches enabling to take advantage of past experiences to improve their decision processes. In this paper, based on Electre tri -a well-known and established MCDA method -, we have presented an approach for identifying the criteria most influencing the decision process regarding the overall performance evaluation of a reduce set of experiences. A case study applying the approach to the humanitarian sector has also been proposed. Such an approach will further be used to enable searching for recommendations in large and highly dimensional search spaces. Future works are envisaged. First, as mentioned above a robustness analysis should be made. Furthermore, we plan to present an extended application, and to sophisticate the proposed model on several aspects: (i) to include veto thresholds into the procedure used to infer model parameters from assignment examples; (ii) to take into consideration the possible interactions/dependences between criteria; (iii) to consider potential disagreements of DMs. Finally, we are also interested by improving the procedure for minimizing the DM's cognitive load (i.e. information requirement), and to both manage and reduce experts' subjectivity.

  ) where α j (a, b) = gj (a)-gj (b)+pj (gj (b)) pj (gj (b))-qj (gj (b)) . Criteria in F c 3 (a, b) vote without reservation for alternative a; in F c 2 (a, b) only the proportion w j • c j is considered voting for alternative a.

  is the one guaranteeing c I (a, b r ) > βλ(a, b r ), β ∈]0, 1]. The criteria of I are those having a large voting weight, i.e. c {j} (a, b r ), and the larger β is, the stronger the contribution of I is. Moreover the individual contribution of each criterion of I should exceed a minimal value γ ∈]0, 1]. This threshold prevents the selection of criteria that could contribute to state c I (a, b r ) > βλ(a, b r ) but in an insignificant proportion. Since the aim is to minimize the subset of criteria to analyse, we have to focus on the smallest subsets I ⊆ F such that the previous conditions are verified. Finally, we can set: I * = arg min I⊆F {|I| : c I (a, b r ) > β * λ(a, b r ) and ∀j ∈ I, c {j} (a, b r ) > γ} (8) where β * is the biggest β ∈]0, 1] such that {I ⊆ F : c I (a, b r ) > β * λ(a, b r )} = ∅ and γ is chosen by the DMs. In case of existence of several subsets I * , the choice of I * is made considering performance of a on the criteria of I * .

  c {1,2,3,4} (a, b 1 ) = c {2,3,4,8} (a, b 1 ) = 0.88 or c {2,3,4,5} (a, b 1 ) = c {2,3,4,7} (a, b 1 ) = 0.89. If a small threshold γ small = c(a,b1) 2 n

Table 1 .

 1 Evaluation criteria.

	Criterion	Preference
	g1 Deadline
	g2 Achievement %
	g3 Portion of the logistics costs
	g4 Human resources
	g5 Added value for the organization
	g6 Added value for the beneficiary country
	g7 Environmental impact
	g8 Security level

Table 2 .

 2 Training set.

	g1 g2 g3 g4 g5 g6 g7 g8 Category
	a1 5 90 38 30 3 4 3 3	C3
	a2 10 97 30 37 3 3 3 4	C3
	a3 12 90 40 50 4 2 4 4	C3
	a4 23 85 40 60 2 2 3 2	C2
	a5 10 74 48 55 3 3 2 3	C2
	a6 20 60 50 65 3 2 3 2	C2
	a7 16 80 55 80 2 2 2 3	C2
	a8 23 55 70 120 1 2 2 1	C1
	a9 60 55 60 125 2 1 1 2	C1
	a10 27 40 50 100 2 3 2 1	C1

Table 3 .

 3 Parameters values obtained from the optimisation problem resolution and given veto thresholds.

		g1 g2 g3 g4 g5 g6 g7 g8
	gj(b1)	26 62 53 80 2 1 2 2
	gj(b2)

Table 4 .

 4 Individual criteria contribution.

		g1	g2	g3	g4	g5	g6	g7	g8
	gj(a)	22	88	52	30	2	4	2	3
	c {j} (a, b1) 0.03 0.46 0.17 0.22 0.038 0.006 0.04 0.03

This de facto prevents the use of traditional Machine Learning approaches.

Note that Electre tri non-compensatory behaviour is defined by the fact that whenever gj(a) -gj(b) is greater than pj(gj(a)) no distinction is made computing the concordance index; a big difference thus cannot compensate any negative difference on another criterion j with j = i.


Abdelhak Imoussaten

Pierre Couturier
email: pierre.couturier@mines-ales.fr

Jacky Montmain

Keywords: multiple criteria decision analysis, fuzzy measure, Choquet integral, preference modelling, difficulty modelling

published or not. The documents may come

Introducing the difficulty of implementing alternatives in the multiple criteria decision problems

Introducing the difficulty of implementing alternatives in the multiple criteria decision problems

1 st Abdelhak Imoussaten I. INTRODUCTION
The aim of Multiple Criteria Decision Analysis (MCDA) methods is to facilitate to the decision-maker (DM) the difficult task of comparing a set of alternatives when several point of view (criteria) should be tacking into account. In this work we focus on the MCDA methods derived from Multi-Attribute Value Theory (MAVT) that are based on a scoring procedure using a value function representing the DM preferences.

More precisely, let us consider a set of alternatives A = {a 1 , a 2 , . . . , a m } and a set of attributes N . For each i ∈ N , the i th attribute measures the extend to which an objective is satisfied and takes its values in a space X i . Each alternative in A is associated with the vector of its evaluation on the attributes of N . To simplify notations, we consider A ⊆ X = i∈N X i . The DM has preferences over elements of X but generally it is difficult to express them. Let us consider that (X, ) represents the preferences of the DM over X where represents a binary relation such that for x, y ∈ X, x y means that the DM prefers x to y. The aim of MAVT [START_REF] Keeney | Decisions with multiple objectives: preferences and value trade-offs[END_REF] is to find a value function v : X → R + that represents these preferences such that: ∀x, y ∈ X, x y ⇔ v(x) ≥ v(y).

(

When the binary relation is a weak order and respects the weak preference independence of the attributes [START_REF] Dyer | Multiattribute utility theory (maut)[END_REF], the authors of [START_REF] Krantz | Additive and polynomial representations[END_REF] state that:

∀x ∈ X, v(x) = F (v 1 (x 1 ), ..., v n (x n )) (2)
where F : R n → R increases in all its arguments and v i : X i → R + is a value function that represents the DM's preferences on attribute i, i ∈ N . The weak preference independence assumption simplifies the assessment of v by assessing v i for a single attribute independently (the other attributes are fixed at arbitrarily selected values) [START_REF] Dyer | Multiattribute utility theory (maut)[END_REF]. An interesting candidate for the operator F is the Choquet integral [START_REF] Labreuche | The choquet integral for the aggregation of interval scales in multicriteria decision making[END_REF] related to a fuzzy measure [START_REF] Murofushi | An interpretation of fuzzy measures and the choquet integral as an integral with respect to a fuzzy Fuzzy sets and Systems[END_REF] that allows modelling interactions between attributes [6] [START_REF] Marichal | Determination of weights of interacting criteria from a reference set[END_REF] and is a generalisation of numerous well-known operators such as the weighted average operator, the ordered weighted average, etc. When v i , i ∈ N and F are known, one can determine a value v(a) for each alternative a ∈ A. Then, we can rank the alternatives in A as follows:

v(a σ(1) ) ≤ v(a σ(2) ) ≤ . . . ≤ v(a σ(m) ) a σ(1) a σ(2) . . . a σ(m)
where σ is a permutation over the set {1, 2, . . . , m}.

The value v(a) quantifies the extend to which the alternative a satisfies all the objectives of the DM. When a has a higher value v(a) than another alternative b this does not signify in all situations that a is more attractive for the DM than b. Indeed, for example, if the implementation of the solution a requires to move from the actual solution a 0 to a and this move is economically or technically more difficult than the move from a 0 to b, b could then be more attractive for the DM. Consequently, we think that the pair constitutes by the satisfaction of an alternative and its difficulty of implementation should be simultaneously taken into account to rank the alternatives of A.

While the economic difficulty analysis decides whether or not ("go" or "no go" decision), the project objectives can be achieved in the given time given the costs constraint, the analysis of technical difficulty is less well defined. It tends to prove that a solution can be developed taking into account the available technology and the resources required. The authors of [START_REF] Bause | Feasibility studies in the product development process[END_REF], distinguish several meanings of the technical difficulty analysis: the analysis of the acceptable or unacceptable consequences of alternatives, the evaluation of the viability of a solution under boundary conditions, the selection among creative ideas of solutions satisfying the operational conditions. In this paper, the meaning of difficulty is close to latest meaning related to operational constraints. The difficulty analysis considered in the paper concerns the quantification of the extend to which it is difficult to move from the actual solution a 0 to an alternative a ∈ A. Few works address the problem of ranking alternatives in a multiple criteria decision analysis problems while integrating into the decision model difficulty considerations. For example, the authors [START_REF] Clivillé | Feasible optimization, ewg mcda 2015[END_REF] consider the difficulty to move between each pair of alternatives in A which requires a large amount of information that may be unavailable.

A more general problem is when the set of alternative A is infinite. This is a multi-objective optimisation problem that consists in finding the optimal improvement of a 0 , maximizing v, while respecting the difficulty constraints. Some works dealing with this problem try to determine which subobjectives to improve first in order to guarantee the maximum gain [START_REF] Labreuche | Determination of the criteria to be improved first in order to improve as much as possible the overall evaluation[END_REF] [11] [START_REF] Montmain | Multicriteria improvement of complex systems[END_REF] [13] [14] [START_REF] Sow | A possibilistic framework for identifying the performance to be improved in the imprecise context of preliminary design stage[END_REF]. In this works the difficulty considered is related either to the cost of improvement or probability/possibility of realisation of such an improvement.

In this paper we propose to introduce the assessment of the alternatives' difficulty in the previous MCDA problems: the problem of ranking alternatives and the problem of improving an actual solution. More, precisely, we propose to represent the difficulty of implementing an alternative using a real function that quantifies the difficulty to move from the actual solution to it. Then we show how to build such a function based on a fuzzy measure.

The paper is organised as follows. In the second section we introduce the difficulty function. Then in the third section we introduce the difficulty function in the ranking problem and in the fourth section we introduce it in the improvement problem. In section five we give an illustration of the proposed approach concerning the problem of improving the results of a student.

II. DIFFICULTY FUNCTION A. Introductory example

To explain the notion of difficulty introduced in this paper, let us start with a simple example concerning a candidate student who wants to improve his chances of being admitted in a prestigious secondary school or university. The student's marks in the first half of the year are presented in Tab. I. The student knows that he should obtain an average greater

B. Single objective difficulty

Let us consider a set of alternatives A included in a space X. Let also consider a starting point x 0 ∈ X. In the same way as for the preference binary relation , we introduce the binary relation x0 where for x, y ∈ A:

x x0 y means that x is more difficult to implement than y starting from x 0 .

If x0 is a weak order over X, we can represent it using a real function, called in this paper difficulty function, d x0 : X → [0, 1] such that:

x x0 y iff d x0 (x) ≥ d x0 (y) (3)
Fig. 1 gives an example of a possible difficulty function that could correspond to the previous example of student candidate. The function shows that the student could obtain easily a score in mathematics which is below 8 = x 0 ; the difficulty increase when he tries to obtain a score higher than 8 until 12; a score above 12 is completely infeasible for him.

Note that the higher is the degree of difficulty of an alternative the lower is the alternative attractive for the DM.

C. Multi-objective difficulty

Let us now consider the case of n objectives measured by n attributes with N = {1, . . . , n}. For each i ∈ N , the i th attribute takes its values in a space X i and we denote X = i∈N X i . Let us consider a vector start x 0 = (x 0 1 , . . . , x 0 n ) ∈ X. Let us consider that (X, x 0 ) represents the difficulty relation over X for the DM. We can state the same equation as [START_REF] Keeney | Decisions with multiple objectives: preferences and value trade-offs[END_REF] to define the difficulty function in the case of multi-objective decision problem. Indeed, the aim is to find a difficulty function d x 0 : X → [0, 1] that represents the difficulty of implementing alternatives in X starting from x 0 such that:

∀x, y ∈ X, x x 0 y ⇔ d x 0 (x) ≥ d x 0 (y). (4)
When the binary relation x 0 is a weak order and respects the weak difficulty independence of the attributes, d x 0 can be stated as follows:

∀x ∈ X, d x 0 (x) = G(d 0 1 (x 1 ), . . . , d 0 n (x n )) (5)
where G : R n → R increases in all its arguments and d 0 i : X i → [0, 1] is the difficulty function related to attribute i starting from x 0 i ∈ X i , ∀i ∈ N . The weak difficulty independence assumption can be stated in the same way as weak preference independence and it simplifies the assessment of d x 0 by assessing d 0 i for a single attribute independently (the other attributes are fixed at arbitrarily selected values). Here also we can consider the Choquet integral as a candidate for the operator G.

Finally, under the previous hypothesis, to determine d x 0 one has just to determine d 0 i , i ∈ N as in subsection II-B and determine the operator G. In section IV, we give two methods to determine d x 0 .

III. INTRODUCING THE DIFFICULTY FUNCTION IN MCDA

PROBLEMS

Let us consider a set of attributes N . For each i ∈ N , the i th attribute takes its values in a space X i and we denote X = i∈N X i . Let also consider a starting vector x 0 . Furthermore, we consider that it exists two functions v : X → R + and d x 0 : X → R + verifying (1) and [START_REF] Labreuche | The choquet integral for the aggregation of interval scales in multicriteria decision making[END_REF]. In this section we consider that v and d x 0 take their values in the interval [0, 1].

A. Ranking alternatives

Let us consider a set of alternatives A = {a 1 , a 2 , . . . , a m } such that A ⊆ X. If one wants to rank the alternatives of A regarding the two objectives v and d x 0 , we are then faced to a multi-objective problem where v has to be maximized and d x 0 has to be minimized. The interesting alternatives for the DM are those belonging to the Pareto optimal set P * such that:

P * = {a ∈ A : b ∈ A such that b dominate a} (6)
where b dominate a means that v(b) ≥ v(a) and d x 0 (b) ≤ d x 0 (a) and at least one of the two inequalities is strict. Obviously, alternatives that reach a high degree of satisfaction require a high degree of difficulty. They are, perhaps, not the best alternatives but they will be in the Pareto optimal set since they are the best related to the first objective.

B. Improving existing solution

The difficulty function can be also introduced in the improvement problems. In particular, we consider the problem of determining the subset of attributes/objectives that one could focus the improvement on, to guarantee a minimal effort for the overall improvement to the greatest extent possible. For this aim works in [START_REF] Grabisch | How to improve acts: an alternative representation of the importance of criteria in mcdm[END_REF] [10] [START_REF] Montmain | Multicriteria improvement of complex systems[END_REF] propose to associate to any coalition of attributes an index named "worth index" that quantifies for any alternative, the average of all the possible expected improvements starting from a fixed point. More precisely, let consider a value function v as in (2

): v(x) = F (v 1 (x 1 ), ..., v n (x n )), ∀x ∈ X where v i : X i → [0, 1].
The worth index for a subset of attributes I ⊆ N , F is considered known, starting from x 0 denoted by w F (x 0 , I) is defined by:

w F (x 0 , I) = 1 0 [F ((1 -τ )v 0 I + τ 1 I , v 0 N \I ) -F (v 0 )] c(v 0 , (1 -τ )v 0 I + τ 1 I , v 0 N \I ) dτ (7) where:
• the vector v 0 = (v 0 1 , . . . , v 0 n ) denotes the vector of partial value functions of the vector x 0 :

v 0 i = v i (x 0 i ), ∀i ∈ N . • for all I ⊆ N , 1 I ∈ [0, 1] |I| is a vector with all components are 1. • for two vectors a, b ∈ [0, 1] n , (a I , b N \I ) ∈ [0, 1] n denotes
the vector with the components of a on attributes in I, and the components of b on the other attributes.

• the quantity

1 0 [F ((1 -τ )v 0 I + τ 1 I , v 0 N \I ) -F (v 0 )] dτ gives the mean value of the gain [F ((1 -τ )v 0 I + τ 1 I , v 0 N \I )-F (v 0 )
] only for improvement vectors on the segment from the current vector values v 0 I (for τ = 0) to the best possible improvement in I, i.e. 1 I (for τ = 1).

• c(v 0 , (1 -τ )v 0 I + τ 1 I , v 0 N \I )
is the cost required for going from vector v 0 to v x 0 ,τ,I which correspond to the cost required for going from alternative x 0 to alternative (1 -τ )v 0 I + τ 1 I , v 0 N \I . In [START_REF] Montmain | Multicriteria improvement of complex systems[END_REF] this cost is considered as not necessary related to monetary considerations, but related to factors they might actually be correlated with risk appraisal, temporal requirements, resources availability, etc. This definition is close to the notion of difficulty introduced in this paper but modelled differently.

When replacing the cost function by the difficulty function, we obtain:

w v (x 0 , I) = 1 0 [F ((1 -τ )v 0 I + τ 1 I , v 0 N \I ) -F (v 0 )] d x 0 (v 0 , (1 -τ )v 0 I + τ 1 I , v 0 N \I ) dτ (8)
The worth index w v (x 0 , I) in ( 8) is a weighted mean of improvements on attributes of I starting from x 0 . To each improvement

[F ((1-τ )v 0 I +τ 1 I , v 0 N \I )-F (v 0 )] is associated a weight 1 d x 0 (v 0 ,(1-τ )v 0 I +τ 1 I ,v 0 N \I )
. The easy improvement will have a high weight then they will contribute with a large gain.

IV. ASSESSING DIFFICULTY FUNCTION

Several techniques could be used to assess the difficulty function in the same way as for the value function in the multiattribute value theory case [START_REF] Keeney | Decisions with multiple objectives: preferences and value trade-offs[END_REF] for additive and multiplicative models. In this section, we present two examples for the case of non-additive model through an aggregation procedure using Choquet integral. The assessment in the first example consists in aggregating partial difficulty functions assuming their existence. Thus, only the fuzzy measure has to be identified. In the second example, we present the disaggregation procedure applied to a non-linear model. In such case, both the partial difficulty functions and the fuzzy measure have to be identified.

In the following we recall the definitions of a fuzzy measure and the Choquet integral.

A fuzzy measure [START_REF] Sugeno | Theory of fuzzy integrals and its applications[END_REF] µ over N is a set function from 2 N to [0, 1] such that: 1) boundary conditions: µ(∅) = 0 and µ(N ) = 1, 2) monotonicity conditions:

∀K, T ⊆ N, K ⊆ T =⇒ µ(K) ≤ µ(T )
Fuzzy measures are necessary to define fuzzy integrals. The most known fuzzy integrals are Choquet integral and Sugeno integral [START_REF] Grabisch | A decade of application of the choquet and sugeno integrals in multi-criteria decision aid[END_REF].

The discrete Choquet integral of an alternative y = (y 1 , ..., y n ) ∈ R +n w.r.t a capacity µ is defined as follows:

C µ (y) = n i=1 (y σ(i) -y σ(i-1) )µ({σ(i), σ(i + 1), ..., σ(n)}) (9)
where 0 = y σ(0) ≤ y σ(1) ≤ y σ(2) ≤ ... ≤ y σ(n) (σ is a permutation over N ).

In the case of difficulty function (9) becomes as follows:

∀x ∈ X, d x 0 (x) = C µ (d 0 1 (x 1 ), . . . , d 0 n (x n ))
A. Direct assessment of a fuzzy measure Let us consider the introductory example of subsection II-A where a student has the scores x 0 i obtained in a first test (see Table II). Suppose that the student should improve his scores to obtain the scores x i , the difficulty values associated are presented in Table II. Suppose that the student wants to use [START_REF] Murofushi | An interpretation of fuzzy measures and the choquet integral as an integral with respect to a fuzzy Fuzzy sets and Systems[END_REF] to assess the difficulty to obtain the score 3 i=1 ω i x i = 12 by improving his first score from x 0 = (8, 10, 12) to x = [START_REF] Montmain | Multicriteria improvement of complex systems[END_REF][START_REF] Couturier | Orienting product performance improvement: application to mechatronics[END_REF][START_REF] Grabisch | A decade of application of the choquet and sugeno integrals in multi-criteria decision aid[END_REF] in the second test. Considering a Choquet integral for G in (5), we get:

d x 0 (x) = C µ d (d 0 1 (x 1 ), . . . , d 0 n (x n ))
where µ d is a fuzzy measure over {m, p, l}.

Suppose that the student tells us that: (r 1 ) it is somewhat easy for him to make an effort that could improve his score in literature, the effort becomes more difficult when it concerns mathematics or physics; (r 2 ) the difficulty change slightly when he has to work only a scientific subject compared to when he has to work literature and a scientific subject and; (r 3 ) the effort is more difficult when he has to work both scientific subjects than only one. From this information, one can deduce the following constraints concerning µ d :

r 1 : µ d ({m}) µ d ({l}) r 1 : µ d ({p}) µ d ({l}) r 2 : µ d ({m, l}) ∼ µ d ({m}) r 2 : µ d ({p, l}) ∼ µ d ({p}) r 3 : µ d ({m, p}) µ d ({m})
r 3 : µ d ({m, p}) µ d ({p})
The following example of µ d (see TABLE III) satisfies the constraints r 1 , r 2 and r 3 . Thus, one can determine the difficulty to obtain the scores [START_REF] Montmain | Multicriteria improvement of complex systems[END_REF][START_REF] Couturier | Orienting product performance improvement: application to mechatronics[END_REF][START_REF] Grabisch | A decade of application of the choquet and sugeno integrals in multi-criteria decision aid[END_REF] from the scores (8, 10, 12): d x 0 (12, 14, 18) = 6. 1 8 .

The fact that the student decides to make the greatest improvement in literature makes the global improvement feasible but requiring a high effort.

B. Disaggregation procedure

A disaggregation procedure [START_REF] Jacquet-Lagreze | Preference disaggregation: 20 years of mcda experience[END_REF] is a procedure which is close to techniques used in machine learning. It assumes a model for the operator to determine, e.g. weighted average, Choquet integral, and then based on information provided by the DM it tries to identify the parameters of the model fitting with this information.

We consider a disaggregation procedure based on MAC-BETH procedure of questioning [START_REF] Costa | On the mathematical foundations of macbeth[END_REF] and its extension to nonadditive model [START_REF] Labreuche | The choquet integral for the aggregation of interval scales in multicriteria decision making[END_REF]. An important issue when aggregating value from several attributes using Choquet integral is to ensure commensurateness. For this aim, in the MACBETH procedure it is assumed that the DM is able to identify for each attribute i two reference levels:

• the reference level 1 d i in X i considered as totally difficult to implement starting from x 0 , even if more difficult elements could exist:

d i (1 d i ) = 1.
• the reference level 0 d i in X i considered as totally easy to implement starting from x 0 , even if more easier elements could exist: d i (0 d i ) = 0. Note that the exponent d in the notations 1 d i and 0 d i is used to make distinction between difficulty and satisfaction. If necessary, in case of satisfaction the notations are 1 v i and

0 v i .
Let us consider a set A of alternatives. Each alternative a ∈ A has n partial scores on n attributes. To determine d x 0 (a) as in [START_REF] Murofushi | An interpretation of fuzzy measures and the choquet integral as an integral with respect to a fuzzy Fuzzy sets and Systems[END_REF], one need to determine µ d and d 0 i (a i ), i ∈ N . To make as easier as possible the comparisons, the DM could be asked to compare the of close alternatives. For instance, on one hand, to obtain µ d , the DM compares alternatives (1 d I ,

0 d I c ) to (1 d J , 0 d J c ), for I, J ⊂ N , i.e. µ d (I) = C µ d (1 d I , 0 d I c
). On the other hand, to obtain d 0 i (a i ), the DM compares alternatives (1 d 1 , ...,

1 d i-1 , a i , 1 d i+1 , ..., 1 d n ) to (1 d 1 , ..., 1 d i-1 , b i , 1 d i+1 , ..., 1 d n ) for i ∈ N , a, b ∈ A.
Note that to facilitate comparisons, several attributes are fixed to the same values in both comparisons.

From these comparisons two linear programs are obtained as in MACBETH procedure where the decision variables are the parameters to determine.

Note that in the case where a human decision-maker provides these preferences, one has to consider small numbers of criteria and alternatives. More then 10 criteria requires comparing at least 55 examples, i.e., alternatives, to expect capturing interactions between criteria.

V. CASE STUDY

Let us consider an illustrative example where two entities play two different roles. On the one hand, the role of candidates is played by different student profiles aiming to integrate a high school. On the other hand, the role of DM is played by two heads of two different high schools. To simplify, we consider only three test scores concerning three different subjects: Mathematics (m); Physics (p) and; Literature (l). Each head of a high school has his own preferences concerning the student profiles. The head of the first high school (HS1) prefers the students with the best general average mark while the head of the second high school (HS2) prefers students with good scientific background but also with a homogeneous profile, i.e., who have not neglected to work literature.

The student who wants to maximize his chance of success to integrate the HS1, will have an interest in working hard on the subjects with the highest coefficients in order to get the best average score. In the contrast, the student who wishes to integrate HS2 will have an interest in working on the subjects where he is weak otherwise a bad score, for example 4/20, in literature is not compensable regarding the preferences of this school.

These two examples of preferences illustrate the idea that, depending on the strategy in place, the improvement decision will not be the same (the optimum will not be the same). Improvement only becomes meaningful when the objectives and priorities are made explicit through a preferences model.

In this case study we propose to help a student to decide which subjects he should improve to guaranty a large gain considering the school that he wants to integrate and his difficulties to improve his scores.

A. Preferences representation

We express the preferences model of HS1 using an additive model considering that mathematics and physics have the same weight and they are more important than literature (see TABLE IV). Concerning HS2, we consider two rules to express its

B. Difficulty representation

Let us now consider that the students did a first test in the middle of the year and they have the second half of the year to improve their bad results. In this case the difficulty of improvement has to be taken into account. Indeed, a student will necessarily considers his ability to do better (or not) in a subject that he has already missed in the first test. Not every improvement represents the same effort for him. If having obtained 08/20 in mathematics is already lucky for a student, he probably has no interest in betting on an improvement in his score in mathematics.

Moreover, if the grade to be raised is 02/20 or 08/20 in a subject that the student knows not to master completely, the challenge is not the same in two different situations: 1) the score 02/20 could appear as a severe sanction which would not reflect his real level and gaining ten points would seem like an accessible objective. 2) the score 08/20 is the true value of the student in the subject throughout the year, it is unlikely that the second test in this subject will allow to gain a lot of points. Thus, the difficulty of the task, the ability to do better, the probability of success, the ratio of profit to effort, also come into play in the choice of what subjects to improve.

Suppose that the scores of the first test are x 0 = (8, 10, 12), an example of the three difficulty functions on each attribute are given in Fig. 2. These functions are Gaussian functions with:

d 0 i (x i ) =      0 if x i ≤ x 0 i exp(- (xi-x e i ) 2 20 ) if x 0 i < x i ≤ x e i 1 if x i > x e i
where x e 1 = 12, x e 2 = 15 and x e 3 = 20 which means that d 0 i increases between x 0 i and x e i and reaches 1, i.e. total difficulty, when the score is 12 in mathematics, 14 in physics and 20 in literature. We consider the fuzzy measure µ d given in TABLE III to aggregate these difficulty functions.

C. The worth index

In this subsection we use the worth index defined in the subsection III-B to determine the subjects that the student should improve first to get the best overall improvement taking into account his difficulties.

Considering the preference model of HS1, we have for I = {m}:

w v (x 0 , {m}) = [ω 1 .(20 -x 0 1 )] 1 0 τ dτ d x 0 (x x 0 ,τ,I ) = 4.5 1 0 τ dτ C µ d (d 0 1 (8.(1-τ )+20.τ ),d 0 2 (10),d 0 3 (12)) = 4.5 µ d ({m}) [ 1/3 0 τ dτ exp(-7.2 (τ -1 3 ) 2 ) + 4 9 ] = 7.2 1/3 0 τ dτ exp(-7.2 (τ -1 3 ) 2 ) + 3.2 = 3.66
In the case of the preference model of HS2, we have:

w v (x 0 , {m}) = 1 0 [C µ v (8.(1-τ )+20.τ,10,12)-C µ v (8,10,12)] C µ d (d 0 1 (8.(1-τ )+20.τ ),d 0 2 (10),d 0 3 (12)) dτ = 1/6 0 3.τ dτ µ d ({m}) exp(-7.2 (τ -1 3 ) 2 ) + 1/3 1/6 [6.τ -0.5] dτ µ d ({m}) exp(-7.2 (τ -1 3 ) 2 ) + 1 1/3 4.5.τ dτ µ d ({m})
= 3.58 This result shows that the student who is in this situation has more interest to improve his score in mathematics when he wants to integrate HS1 than when he wants to integrate HS2. The following tables VI and VII give the worth index of all subsets of N for the two preferences models. As we can see in these two tables, the student obtains the same and large added worth with both preference models if he improves his score in Literature. With the preference model of HS1, the student has a possibility to obtain a better added worth when improving literature than when improving all the subjects. While with the second model of HS2, he has two other possibilities: improving Mathematics and Literature or improving Physics and Literature. The idea of this illustrative example, is to show that the student who has great difficulties in mathematics will not necessarily choose this subject to catch up the bad score of the first test even if the weight of Mathematics is very high.

VI. CONCLUSION

Concerning theoretical aspect, this paper proposes a mathematical models allowing the DM to assess alternatives not only by quantifying their degrees of satisfaction but also by quantifying the degree of difficulty to implement these alternatives. The illustration shows that the student who obtained bad scores in a first test, tends often to work hard on the subjects that he missed. However, depending on the high school that he desires to integrate, his improvement actions should not be the same. Spending a lot of time and money to improve his scores on the subjects that are very hard for him is not always the winning strategy. When a student is aware of his difficulties and of the preferences model that is used to aggregate his scores, the worth index aggregating difficulty functions could help him to decide which subjects guarantee a large gain considering his own difficulties. In the future works, risk and uncertainty considerations can also be taken into account when assessing a difficulty function.

Fig. 1 .

 1 Fig. 1. Example of a difficulty function.

Fig. 2 .

 2 Fig. 2. Difficulty function of the three subjects.

TABLE VI THE

 VI WORTH INDEX WITH THE PREFERENCES MODEL OF HS1

		∅	{m}	{p}	{l}
	wv(x 0 , .)	0	3.66	3.2	17.87
		{m, p}	{m, l} {p, l} {m, p, l}
	wv(x 0 , .)	4.53	4.96	4.61	5.4


domain: chim sdv

text:
Mouna Mokhtari

Sonia Chabani
email: s.chabani@gmail.com

Abdulmagid Alabdul Magid

Mohammed Benkhaled

Laurence Voutquenne-Nazabadioko

Hamada Haba

A new flavonoid glycoside from the aerial parts of Diplotaxis erucoides (L.) DC. growing in Algeria

Keywords: Brassicaceae, Diplotaxis erucoides, Kaempferol-3-O-[α-L-rhamnopyranosyl-(1→2)-β-D-xylopyranoside]-7-O-α-L-rhamnopyranoside, flavonoids, NMR

) and two monoglycerolipids (7, 8). Their structures were established by extensive spectroscopic analysis including 1D-and 2D-NMR

( 1 H, 13 C, 1 H-1 H COSY, HSQC and HMBC), mass spectrometry (HR-ESI-MS) and by comparison with the data reported in the literature.

Introduction

The genus Diplotaxis (wall-rocket), belonging to the Brassicaceae (Cruciferae) family, is represented by more than 32 species (Warwicket al. 2006) which are mainly distributed in the Mediterranean region, Southwest Asia and Europe [START_REF] Pignone | Diplotaxis[END_REF].

Plants of this genus have been used in folk medicine as aphrodisiac and anti-diarrheic (De Feoet al., 1992). They have also mucolytic properties [START_REF] Caudron | Un mucolytique d'origine végétale Diplotaxis tenuifolia[END_REF].

Diplotaxis erucoides (L.) D.C. (white wall-rocket) is an herbaceous plant up to 30-50 cm tall.

Its flowers are white veined with purple [START_REF] Quezel | New flora of Algeria and the southern desert regions[END_REF]. It is a wild vegetable with high content in ascorbic acid, glucosinolates, flavonoids, and it is also a NO3 -bioaccumulator. [START_REF] Bennett | Ontogenicpro filing of glucosinolates, flavonoids, and other secondary metabolites in Erucasativa (salad rocket), Diplotaxis erucoides(wall rocket), Diplotaxis tenuifolia (wild rocket), and Bunias orientalis (turkishrocket)[END_REF][START_REF] Guijarro-Real | Morphological diversity and bioactive compounds in wall rocket (Diplotaxis erucoides (L.) DC.)[END_REF]. Previous phytochemical investigations have reported that n-hexane and n-butanol fractions of the aerial parts of Diplotaxis erucoides contained fatty acid and flavonoids [START_REF] Salah | Phytochemical and biological investigation of two Diplotaxis species growing in Tunisia: D. virgate & D. erucoides[END_REF].

In the present study, we describe the isolation and structural elucidation of one new flavonoid glycoside (1) named diploerucoside A with seven known compounds (2-8) from the aerial parts of D. erucoides. All compounds were identified based on spectroscopic data (1D-and 2D-NMR, HR-ESI-MS) and comparison with data from the literature.

Results and discussion

Compound 1 was isolated as yellow crystals. Its molecular formula was assigned as C32H38O18 on the basis of the high-resolution ESI-MS experiment (negative-ion mode) by 709.1985) which is consistent with the presence of 32 signals on the 13 C-NMR spectrum. The analysis of the 1 H-and 13 C-NMR spectra of 1 showed characteristic signals of a glycosylated flavonoid. The 1 H NMR and 1 H-1 H COSY spectra displayed two meta coupled signals at δH 6.46 (1H, d, J = 1.5 Hz, H-6) and 6.75 (1H, d, J = 1.5 Hz, H-8), characteristic of the 5,7-distributed A ring of flavonoid. These protons were resonating over 6.4 ppm indicating that glycosylation site is position 7 [START_REF] Vassallo | New flavonoid glycosides from Chrozophora senegalensis and their antioxidant activity[END_REF]. The 1 H NMR spectrum also showed two ortho-coupled doublet signals at δH 8.06 (2H, d, J= 8.5 Hz) and 6.91 (2H, d, J= 8.5 Hz) of a para-substituted aromatic nucleus. According to the 1 H-1 H COSY spectrum, these protons belong to the same spins system, attributable to the H-2'/H-6' and H-3'/H-5' protons. Their carbons were detected by HSQC spectrum at δC 116.2 and 132.2, respectively. The assignments of all carbons of the flavonol moiety were accomplished on the basis of 3 JH-C HMBC correlations depicted between H-2′/H-6′ and two quaternary carbons C-2 (δC 159.1) and C-4' (δC 161.7); H-3'/H-5' and the quaternary carbon at δC 122.7 (C-1'); H-8 and four carbons, among them three are quaternary at δC 100.5 (C-6), 107.5 (C-10), 158 (C-9) and 163.5 (C-7); H-6 and C-5 (δC 163), C-8 (δC 95.5). Thus, the aglycone part of 1 was identified as kaempferol [START_REF] Aisyah | Flavonoid compounds from the leaves of Kalanchoe prolifera and their cytotoxic activity against P-388 murine leukemia cells[END_REF]. Furthermore, three anomeric proton signals were clearly observed at δH 5.63 (d, J = 7 Hz), 5.22 (brs) and 5.56 (brs) correlating with carbon signals at δC 101.2, 99.9 and 102.7 in the HSQC spectrum, respectively. The analysis of 1 H-1 H COSY and HSQC experiments, and acid hydrolysis led to the identification of β-D-xylose and two α-L-rhamnose units. In fact, the two α-L-rhamnopyranosyl units with anomeric proton signals at δH 5.56 (brs, H-1"") and 5.22 (brs, H-1‴) were characterized by the presence of eight proton spins system with their methyl groups at δH1.33 (d, J = 6.2 Hz, H3-6"") and 1.12 (d, J = 6 Hz, H3-6‴), respectively. Additionally, the β-D-xylopyranosyl unit was assigned on the basis of six proton spins system starting from the third anomeric proton H-1" (δH 5.63). This latter (H-1") correlated with H-2" (δH 3.66, dd, J = 9.5, 7 Hz) in the 1 H-1 H COSY spectrum, which coupled with H-3" (δH 3.51, t, J = 9.5 Hz). The proton H-4" resonating as a triplet at δH 3.49 (J = 9.5 Hz) correlated with the protons H2-5" at δH 3.74 (dd, J = 11.5, 5 Hz) and 3.1 (dd, J = 11.5, 7.8 Hz) [START_REF] Serge | Chemical composition and anti-herpes simplex virus type 1 (HSV-1) activity of extracts from Cornus canadensis[END_REF]. The values of the coupling constants are all greater than 7 Hz indicating a trans-diaxial position of these protons. The correlation observed in the HMBC spectrum (Figure 1) from anomeric proton H-1"" (δH 5.56) of rhamnose to carbon C-7 (δH 163.5) revealed the point of attachment of one of the α-L-rhamnopyranosyl units to the aglycone part at C-7 position. On the other hand, the long-range correlations (Figure 1) observed between the anomeric proton H-1‴ (δH 5.22) of the second rhamnose with C-2″ (δC 79.5) of xylose indicated the interglycosidic linkage between these glycosyl units identified as α-L-rhamnopyranosyl-(1→2)-β-D-xylopyranoside. Moreover, the glycosylation position of this diglycoside (Figure 1) was deduced into C-3 based on the chemical shift of this carbon at δC 134.7 and C-2 at δC 159.1 [START_REF] Chabani | Flavonoid glycosides and triterpenoids from Atractylis flava[END_REF][START_REF] Okoye | Flavonoid glycosides from Olax mannii: Structure elucidation and effect on the nuclear factor kappa B pathway[END_REF]. Thus, 1 was

observation of pseudo-molecular ion peak at m/z = 709.1985 [M-H] -(calcd [C32H37O18] -,
determined to be kaempferol-3-O-[α-L-rhamnopyranosyl-(1→2)-β-D-xylopyranoside]-7-O-α- L-rhamnopyranoside, a new flavonoid glycoside named diploerucoside A.

Insert figure 1 here

Compared the physical and spectral data (1D-and 2D-NMR, HR-ESI-MS) with the literature data, 7 known compounds (2-7) were identified as one flavonoid: astragalin (2) [START_REF] Fawzi | Isolation of astragalin from Cressacretica cultivated in Iraq[END_REF][START_REF] Ghavam-Haghi | Isolation and identification of astragalin and 2-methoxy tyrosol from the bulbs of Allium paradoxum[END_REF]; one phenolic glucoside: 1-O-p-coumaroyl-β-D-glucose (3) [START_REF] Strack | Structures and accumulation patterns of soluble and insoluble phenolics from norway spruce needles[END_REF][START_REF] Baderschneider | Isolation and characterization of novel benzoates, cinnamates, flavonoids, and lignans from Riesling wine and screening for antioxidant activity[END_REF]; one monoterpene: loliolide (4) [START_REF] Hodges | The structure of Loliolide: A terpene from Lolium perenne[END_REF]; one triterpene: lupeol (5) [START_REF] Saratha | Isolation and characterization of lupeol, a triterpenoid from calotropis gigantea latex[END_REF]; β-sitosterol (6) (Nyigo et al., 2012), and two monoglycerolipids: (10E,12Z)-2,3-dihydroxypropyl octadeca-10,12-dienoate (7) [START_REF] Eleni | High resolution NMR spectroscopy as a structural and analytical tool for unsaturated lipids in solution[END_REF], and (9Z,11E)-2,3-dihydroxypropyl octadeca-9,11-dienoate (8) [START_REF] Chin | Dietary Sources of Conjugated Dienoic Isomers of linoleic acid, a newly recognized class of anticarcinogens[END_REF][START_REF] Eleni | High resolution NMR spectroscopy as a structural and analytical tool for unsaturated lipids in solution[END_REF] (Figure 2). Compounds 3, 4, 7 and 8 were isolated from Diplotaxis genus for the first time.

Insert figure 2 here

This study allowed particularly the isolation and identification of two flavonoids named diploerucoside A (1) and astragalin (2). This type of secondary metabolites is used as a chemotaxonomic marker in genus Diplotaxis including D. erucoides, D. acris, D. harra and D. muralis [START_REF] Hussein | Chemosystematic significance of flavonoids isolated from Diplotaxis acris (Brassicaceae) and related taxa[END_REF]. The new flavonol glycoside diploerucoside A (1) is a kaempferol derivative having two sugar moieties in positions C-3 (α-L-rhamnopyranosyl-(1→2)-β-D-xylopyranoside) and C-7 (α-L-rhamnopyranoside), whereas the known flavonol

(2) possesses a kaempferol skeleton with β-D-glucose in position C-3. Furthermore, previous studies revealed that D. erucoides contained flavonol glycosides with kaempferol, quercetin and rhamnetin skeletons and that glycosylation linkage occurred mainly at positions C-3 and C-7 of the aglycone (Ben [START_REF] Salah | Phytochemical and biological investigation of two Diplotaxis species growing in Tunisia: D. virgate & D. erucoides[END_REF]Sanchez-Yelamo, 1994). This constatation is notably in agreement with our results helping to further clarify the involvement of flavonoids in the chemotaxonomic profile of Diplotaxis species.

The Experimental

General experimental procedures

The 1D-( 1 H and 13 C) and 2D-( 1 H-1 H COSY, HSQC, HMBC) NMR spectra were recorded on a Brüker type device at 500 MHz for 1 H and 125 MHz for 13 cm)] plates, detection at 254 and 366 nm and by spraying with sulfuric acid reagent (50%) followed by heating.

Plant material

The aerial parts of Diplotaxis erucoides (L.) DC. were harvested in November 2018 in El Madher from region of Batna (North eastern of Algeria). The plant was identified by Prof.

Bachir Oudjehih from the Agronomy Department of the Institute of Veterinary and Agronomic Sciences at University of Batna-1 (Algeria). A voucher specimen was listed under the number 977/LCCE.

Extraction and isolation

Powdered air-dried aerial parts (1.4 kg) of Diplotaxis erucoides were extracted with 70% aqueous ethanol for 48 h at room temperature. The extraction was repeated twice with the renewal of the solvent. After filtration and concentration under vacuum, the concentrated aqueous residue (400 mL) was submitted to liquidliquid partition, with petroleum ether (PE), ethyl acetate (EtOAc), and n-butanol (n-BuOH), successively (each solvent, 250 mL × 3). The evaporation of the organic phases gave 15.59 g of PE, 14.17 g of EtOAc and 18.48 g of n-BuOH fractions.

A part of the EtOAc fraction (7 g) was separated by VLC performed over a silica gel (5 × 5 cm; fractions of 100 mL). The elution was carried out with PE/EtOAc (10:0 to 0:10), then EtOAc/MeOH (10:0 to 0:10) to yield fractions (F1-F9). Fraction F2 (28 mg) was fractionated over a silica gel CC, eluted with PE/EtOAc (10:0 to 0:10) to give seventeen subfractions. The sub-fractions F2-5 (6.1 mg) and F2-7 (5.7 mg) were subjected successively to Sephadex LH-20 column, eluted with CHCl3 to provide compounds 5 (3 mg) and 6 (3 mg).

Fraction F5 (88.2 mg) was purified by silica gel CC eluted with a gradient of PE/EtOAc (10:0 to 0:10) to give four sub-fractions (F5-1 -F5-4). Compounds 4 (2.8 mg), 7 (2.5 mg) and 8 (2 mg) were obtained from sub-fraction F5-3 (42.6 mg) using Sephadex LH-20 CC, eluted with CH2Cl2 followed by silica gel preparative thin layer chromatography (prep. TLC) eluted with CH2Cl2/MeOH (9:1).

The n-BuOH extract (7 g) was subjected to a silica gel VLC, eluted with CHCl3/MeOH/H2O solvent system to give eight fractions. Fraction F4 (1.292 g) was subjected to a polyamide SC-6 CC eluted with the H2O/MeOH system (10:0 to 0:10). Ten sub-fractions were obtained after fractionation and pooling (A1 to A10). The A4 sub-fraction (106.2 mg) was separated on Sephadex LH-20 CC, eluted with MeOH, followed by silica gel CC, eluted with CHCl3/MeOH (10:0 to 5:5), to afford eight sub-fractions (A4-1 to A4-8). Compound 3 (1.5 mg) was obtained by precipitation of the sub-fraction A4-4 (11.4 mg) in MeOH. The sub-A4-7 (6.7 mg) was precipitated in diethyl ether followed by filtration through a Sephadex LH-20 CC using MeOH as eluent to provide compound 1 (6.4 mg). Sub-fraction A4-8 (62.9 mg) was chromatographed on Sephadex LH-20 CC, eluted with CHCl3/MeOH to yield 9 sub-fractions (A4-8a to A4-8H). The sub-fraction A4-8f (11 mg) was purified by silica gel prep. TLC in (CHCl3/MeOH/H2O; 7:3:0.3) to furnish compound 2 (3.2 mg).

Acid hydrolysis

An acid hydrolysis was carried out on compound 1 to establish the configuration of monosaccharide residues. Indeed, 5 mg of 1 was refluxed with 5 mL of 2N HCl for 2h, then it was neutralized with 0.5 M KOH. After extraction with CH2Cl2 (3 × 5 mL), the aqueous solution was evaporated and then purified by prep. TLC using the solvents system MeCOEt-isoPrOH-Me2CO-H2O (10:5:

Spectral data

Kaempferol-3-O-[α-L-rhamnopyranosyl-(1→2)-β-D-xylopyranoside]-7-O-α-L-rhamnopyrano- side (

Conclusion

A new flavonol glycoside (1) named diploerucoside A along with seven known compounds including one flavonoid, one phenolic glucoside, one monoterpene, two monoglycerolipids, one sitosterol and one triterpene have been isolated from the EtOAc and n-BuOH fractions of

  C. Samples were dissolved in CD3OD or CDCl3 deuterated solvents depending on the solubility of the compounds. The chemical shifts (δ) are expressed in ppm relative to tetramethylsilane (TMS); the coupling constants are defined in Hz. HR-ESI-MS experiments were performed on a Waters SYNAPT G2-Si High Resolution Q-TOF Mass Spectrometer equipped with electrospray ionization (ESI) source. The optical rotation [α]D of the isolated compounds was determined using a Perkin-Elmer 241 polarimeter type at 20 °C. The fractionation was carried out on vacuum liquid chromatography (VLC) using SiO2 (Kieselgel Merck 70-230 mesh). The purification of isolated compounds was performed over column chromatography (CC) on SiO2 (Kieselgel Merck 70-230 mesh), polyamide SC6 and Sephadex LH-20 (25-100 μM). Analytical and preparative TLCs were conducted on Silica gel [Kiselgel 60 F254, Merck 250 μM (20 × 20

  3.5) to afford two monosaccharides identified as rhamnose and xylose by comparison with authentic simples on TLC in the same solvents system. The optical rotation of each purified monosaccharide was measured [rhamnose [α] D 20 +2.4 (c 0.96, H2O) and xylose [α] D 20 +15.3 (c 0.96, H2O)] and the absolute configurations were determined as L for rhamnose and D for xylose by comparison with [α] 20 D values of authentic samples of D-xylose and L-rhamnose (Sigma-Aldrich).

  1): yellow crystals; mp 252-253 °C; [α]D 20 -21.6 (c 0.15, MeOH); HR-ESI-MS = 709.1985 [M -H] -(calcd [C32H37O18] -, 709.1985); 1 H-NMR (500 MHz, CD3OD) δH 8.06(2H, d, J = 8.5 Hz, H-2′/H-6′), 6.91 (2H, d, J = 8.5 Hz, H-3′/H-5′), 6.75 (1H, d, J = 1.5 Hz, H-8), 6.46 (1H, d, J = 1.5 Hz, H-6), 5.63 (1H, d, J = 7 Hz, H-1″), 3.66 (1H, dd, J =9.5, 7 Hz, H-2″), 3.51 (1H, t, J = 9.5 Hz, H-3″), 3.49 (1H, t, J = 9.5 Hz, H-4″), 3.74 (1H, dd, J = 11.5, 5Hz, H-5″a), 3.1 (1H,dd, J = 11.5, 7.8 Hz, H-5″b), 5.22 (1H, brs, H-1‴), 4.01 (1H, dd, J = 3.5, 1.2 Hz, H-2‴), 3.77 (1H, dd, J = 9.5, 3.5 Hz, H-3‴),3.36 (1H, t, J = 9.5 Hz, H-4‴), 4.06 (1H, dd, J = 9.8, 6 Hz, H-5‴), 1.26 (3H, d, J = 6 Hz, H3-6‴),5.56 (1 H, brs, H-1″″), 4.02 (1H, dd, J = 3.4, 1.1 Hz, H-2″″), 3.83 (1H, dd, J = 9.5, 3.4 Hz, H-3″″), 3.48 (1H, t, J = 9.5 Hz, H-4″″), 3.6 (1H, dd, J = 10.2, 6.4 Hz, H-5″″), 1.04 (3H, d, J =6.5 Hz, H3-6″″).13 C-NMR (125 MHz, CD3OD) δC 179.0 (C-4), 163.5 (C-7), 163.0 (C-5), 161.7 (C-4′), 159.1 (C-2), 159.0 (C-9), 134.7 (C-3), 132.2 (C-2′/C-6′), 122.7 (C-1′), 116.2 (C-3′/C-5′), 107.5 (C-10), 100.5 (C-6), 95.5 (C-8), 101.2 (C1″), 79.5 (C-2″), 77.5 (C-3″),71.4 (C-4″), 67.2 (C-5″), 102.7 (C-1‴), 72.4(C-2‴), 72.3(C-3‴), 74.0 (C-4‴), 70.0 (C-5‴), 17.7 (C-6‴), 99.9 (C-1″″), 71.7 (C-2″″), 72.1 (C-3″″),73.6 (C-4″″), 71.3 (C-5″″), 18.1 (C-6″″).

Figure 1 .Figure 2 .

 12 Figure 1. HMBC (H-C) correlations for compound 1

Acknowledgement

We would like to thank the DGRSDT and the Ministry of Higher Education and Scientific Research of Algeria for PRFU (B00L01UN050120180001),Conseil General de la Marne, Ministry of Higher Education and Research (MESR) France and EU-program FEDER to the PlAneT CPER project (University Reims Champagne-Ardenne).

Disclosure statement

No potential conflict of interest was reported by the authors.

the 70% ethanol extract of the aerial parts of Diplotaxis erucoides belonging to the Brassicaceae family.This study reports for the first time the occurrence of compounds 3, 4, 7 and 8 in the Diplotaxis genus.

Supplementary material

HR-ESI-MS, 1 H and 13 C NMR, 1 H-1 H-COSY, HSQC and HMBC spectra of all compoundsare available online.


Roberto Bubbico
email: roberto.bubbico@uniroma1.it

Gilles Dusserre

Barbara Mazzarotta

Calculation of the Flame Size from Burning Liquid Pools

The calculation of the consequences associated with a pool fire consists in a stepwise procedure where a number of parameters must be characterized, which depend on the geometrical size and shape of the flame. A number of calculation models exists in the literature, characterized by different levels of accuracy and complexity.

In the present work we will focus on the characterization of the geometrical configuration of the flame generated from a pool fire: some of the most commonly adopted models will be shortly recalled, and compared against experimental data taken from the literature. It is expected that this would provide useful information about the range of applicability and the level of accuracy of these models. Also, it will help improve the quality of the results, and reduce the time required for carrying out important applications such as consequence assessment and risk analysis, where a large number of calculations must be run.

Introduction

Pool fire represents one of the most common accidental events which can occur within an industrial installation [START_REF] Vilchez | Historical analysis of accidents in chemical plants and in the transportation of hazardous materials[END_REF], however, since it is generated by the ignition of the release of any flammable liquids, it can actually happen in a wide range of situations, like during the transportation of traditional liquid fuels from storage or production sites to the final destination, and in other everyday-life conditions. Unless a very large release has occurred, the consequences of such an event are usually limited to a relatively small area surrounding the location of the leak, compared with other more harmful accidents characteristic of the process and power industry, like VCEs and fireballs. Nonetheless the estimation of the impact area of pool fires is of great importance, especially when the possibility of generating accidents characterized by larger impact areas and more serious consequences (domino effect), is present. The damages caused by a liquid fire are mainly connected with the heat radiated in the surrounding area, which depends on a number of parameters: the energy per unit time emitted by the flame, the distance from the fire, the characteristics of the target surface receiving the radiated heat (e.g. shape and size), and a number of other environmental conditions (atmospheric transmissivity, wind intensity, etc.). At least in theory, the heat received by a given target might be calculated by means of the classical Stefan-Boltzmann equation, but due to the wide range of variability of the involved parameters, and to the uncertainty in their quantitative assessment, a number of theoretical and semi-empirical models have been proposed over the years and are still commonly adopted by safety engineers. Since they do not require the knowledge of many specific input parameters, another advantage of these correlations, lies in the fact that they can be easily applied in a number of applications where a great amount of quick calculations have to be carried out, such as in Risk Analysis. Nonetheless, because of the lack of experimental data, uncertainties are still present about their practical applicability. In this view, and with reference to other dangerous phenomena like explosions, it has already been shown that an assessment of the reliability and of the range of applicability of these models is of the higher importance [START_REF] Bubbico | Analysis and comparison of calculation methods for physical explosions of compressed gases[END_REF]. Consequently, a similar analysis will be repeated here with reference to some of the most widely known and adopted pool fire models.

Literature background

Assessing the damages caused by a pool fire is a stepwise procedure: following a backward sequence it can be observed that the heat received by a given target depends on a number of factors, such as the heat actually radiated by the fire, the fraction of heat "lost" along the travel from the emitting (the flame) to the receiving surface, the distance and the relative exposure of the receiving surface with respect to the emitting one (all of them summarized in the so-called geometric view factor). Most of these factors depend either on the geometrical size and shape of the flame and/or on the heat produced by the fire. The geometry of the fire is in turn a function of different parameters: some of them are intrinsic with the fuel involved in the fire (e.g. the physical properties, the heat of combustion, etc.); other parameters are somehow accidental, like the release rate, the size of the pool at the base of the fire (whether confined or not), the properties of the surface on which the liquid pool is formed (physical and thermal properties, whether flat or inclined terrain, etc.), the wind velocity and direction, and many others. Similarly, the heat produced by the fire is a function of the combustion rate and the heat of combustion of the fuel. As might be expected, for each of the above parameters a number of calculation models exist, characterized by different levels of accuracy and complexity. Frequently, a higher accuracy is associated with a higher complexity of the model, which, in turn, often implies a larger number of input parameters for the model and longer calculation time. A higher accuracy is usually required when a specific case is under investigation, for example when at least some the parameters involved are already known in advance (e.g. pool size, wind velocity, distance from the target, etc.). In other cases, a lower accuracy can be tolerated, provided that shorter calculation times are obtained; this is the case of risk analysis, for example, where all the possible accidental scenarios have to be taken into consideration and simulated (different pool sizes, fuel and/or ambient temperatures, wind velocity and direction, distances, and so on). In these cases, an accurate analysis of all these scenarios would require an unacceptable calculation time; as a consequence, an optimum between results accuracy and calculation time is here sought, and a number of simplified models and correlations are thus available in the literature. As stated above, the present work will focus on the characterization of the geometrical configuration of the flame generated from a pool fire: some of the models most commonly adopted in risk analysis will be shortly recalled in this section, and they will then be compared against experimental data obtained from the literature in the next section.

Burning rate

The first step in calculating the consequences of a pool fire consists in the calculation of the burning rate, that is the velocity of consumption of the fuel from the liquid pool. Three of the most common models will be used here. [START_REF] Zabetakis | Research on the Hazards Associated with the Production and Handling of Liquid Hydrogen[END_REF] proposed the following relation for the calculation of the burning rate from hydrocarbon liquid pools:

" = " 1 - (1)
where m" ∞ is the specific mass burning rate at "infinite" diameter (kg/m ² /s) to be experimentally determined, k is the absorption-extinction coefficient of the flame (m -1 ), β is a correction coefficient for the beam length, and D is the pool diameter (m). According to this equation, the pool diameter has an explicit influence on the burning rate of the fuel, at least at low diameter values. A significant drawback of this equation is that all parameters are fuel-dependent and have to be determined experimentally: reliable measurements are available in the literature for a few substances, only, and in some cases different values are provided for the same fuels [START_REF] Babrauskas | Estimating Large Pool Fire Burning Rates[END_REF][START_REF] Rew | Development of pool fire thermal radiation model[END_REF][START_REF] Munoz | Analysis of the geometric and radiative characteristics of hydrocarbon pool fires[END_REF]. [START_REF] Babrauskas | Estimating Large Pool Fire Burning Rates[END_REF] also suggests deriving the combination kβ and not separated into k and β. At larger scales (i.e. around 1 m diameter), the influence of the pool diameter vanishes and [START_REF] Burgess | Diffusive Burning of Liquid Fuels in Open Trays[END_REF] correlated the burning rate of a number of liquid fuels (specifically, liquid hydrogen, liquid natural gas, butane, methanol, hexane, benzene, and xylene) by the simple relationship:

" = 1.27 • 10 ∆ ∆ * (2)
where ρ L is the density of liquid fuel at boiling point, ΔH c is the heat of combustion (kJ/kg), and ΔH* v is the modified heat of vaporization at the boiling point of the liquid given by

∆ * = ∆ + (3).
In Eq(3) ΔH v is the heat of vaporization of the liquid at the boiling temperature (kJ/kg) and c p is the heat capacity of the liquid (kJ/(kg°C)). Eq(2) has been recognized as not accurate enough for liquefied gases [START_REF] Mudan | Thermal radiation hazards from hydrocarbon pool fires[END_REF], and a more general, though less accurate, fitting equation has been proposed:

" = 1 • 10 ∆ ∆ * (4).
An advantage of equations ( 2) and ( 4), with respect to Eq(1), is that they only need the heat of vaporization and heat of combustion of the involved fuel, which are much more easily available than the experimental parameters required by Eq(1). Many other theoretical or empirical correlations are available in the literature (see for example [START_REF] Joulain | The behavior of pool fires: state of the art and new insights[END_REF][START_REF] Ris | Radiation fire modeling[END_REF][START_REF] Raj | Large hydrocarbon fuel pool fires: Physical characteristics and thermal emission variations with height[END_REF], however they will be not adopted here either because of their limited range of validity or because of the time required to get the input data or to run the calculations, which are not always acceptable during a full risk analysis.

Flame height

The flame height is a fundamental parameter to assess the consequences associated with a pool fire, and a large number of models have been proposed over the years for its quantification (just to name a few, see [START_REF] Rew | Modelling of thermal radiation from external hydrocarbon pool fires[END_REF][START_REF] Mudan | Calculating impacts for large open hydrocarbon fires[END_REF][START_REF] Mudan | Thermal radiation hazards from hydrocarbon pool fires[END_REF][START_REF] Raj | Large hydrocarbon fuel pool fires: Physical characteristics and thermal emission variations with height[END_REF][START_REF] Munoz | Predicting the emissive power of hydrocarbon pool fires[END_REF]. The best known and most widely adopted correlation for calculating the ratio between the flame height and the diameter of a circular pool is the one by [START_REF] Thomas | The Size of Flames From Natural Fires, 9th International Symposium on Combustion[END_REF]

: = 42 • " .
(

) 5
where m" is the specific mass burning rate (kg/m ² /s) calculated by Eq(2), ρ a is the air density (kg/m 3 ), and g the gravitational acceleration (m/s 2 ). Thomas also suggested an improved correlation to take into account the influence of the wind, however the value of the wind speed is rarely known from the available experimentations, and therefore this version will not be considered here. Other correlations have been put forward by different Authors, however they are either specific for particular fuels (see for example [START_REF] Moorhouse | Scaling Criteria for Pool Fires Derived From Large-Scale Experiments[END_REF], or they proved to be less accurate than Eq(5) [START_REF] Mudan | Thermal radiation hazards from hydrocarbon pool fires[END_REF]. Based on a variety of experimental measurements (even from jet fires), [START_REF] Heskestad | Fire Plumes, Flame Height, and Air Entrainment[END_REF] proposed a correlation where the height of the flame is dependent on the heat produced by the fire, instead of the burning rate as in Eq(5):

= .

-1.02 (6) with Q (kW) the heat release rate. Differently from Eq(5) and Eq(6), where the size of the flame is a function of the burning conditions, represented by either the mass burning rate m" or the heat release rate Q, the BrLtz correlation [START_REF] Zhang | Accident consequence simulation analysis of pool fire in fire dike[END_REF] represents an even simpler approach, since the height of the flame is dependent on the diameter of the pool only:

= 1.73 + 0.33 . (7).

Results and discussion

The above equations have been applied to a number of experimental data obtained from the literature. Unfortunately literature data are scarce in quantity and often incomplete (missing information on weather data, like temperature and wind velocity, on the physical properties of the fuels, and so on). As a consequence it is usual making reference to well known and often referred to experimental campaigns. Some of these data have already been summarized in previous papers, so, for the sake of simplicity, the data used in the present work have been directly derived from them, and in particular from [START_REF] Babrauskas | Estimating Large Pool Fire Burning Rates[END_REF] and [START_REF] Rew | Development of pool fire thermal radiation model[END_REF].

Burning rate

In order to check the accuracy of the above models, in Figure 1 the experimental values of the burning rate of hexane pool fires are reported along with the calculations provided by Eqs (1), ( 2) and (4). It is apparent that, while Eqs ( 2) and ( 4) provide rather good (constant) values of the infinite diameter burning rates, only Eq (1) is able to predict accurate values of the burning rate in the whole range of diameters investigated, and in particular for small pools (i.e., in the case of hexane, for D less than about 1.5 m). The same behaviour is observed for most of common hydrocarbons and conventional liquid fuels (e.g., heptane, gasoline, JP4, etc.). Conversely, none of the adopted equations is able to reliably predict the burning rate for liquefied gases (LNG and LPG), as shown in Figure 2. This is in agreement with similar observations by [START_REF] Babrauskas | Estimating Large Pool Fire Burning Rates[END_REF]. The above results demonstrated that Eq (1) provides better estimates of the burning rate of most hydrocarbons, with respect to Eqs (2) and (4). However, two considerations should be highlighted: the use of Eq(1) requires the knowledge of specific parameters strictly linked to the investigated fuel, which have to be experimentally determined and are not easily found in the literature; furthermore, from a practical point of view, in many cases pool fires generated by relatively large pools are of interest (which often means larger than 1 m), and in these cases the infinite diameter burning rates could suffice. On the other hand, it is worth saying that even in the case of Eqs ( 2) and ( 4), when mixtures of hydrocarbons are involved, which are not always very well characterized in terms of composition, their physical properties, such as the heat of evaporation and the heat of combustion which appear in Eq (3), cannot be accurately calculated, and consequently a significant uncertainty can be found in the estimation of the mixture burning rate.

Flame height

In Figure 3 the experimental values of the flame height of hexane and heptane pool fires are reported along with the values calculated with Eqs(5) to ( 7); the curves are on a log-log scale for a better understanding of the trends. In the case of Eq(6), the heat production rate is required, which is linked to the burning rate of the fuel; since, as shown in section 2.1, this latter parameter can be calculated with different equations, different

Exp

Eq (1) Eq (2) Eq (4)

versions of Eq(6) can be correspondingly derived. In the present case Eqs(1) and ( 4) have been adopted, giving rise to Eqs(6a) and (6b), respectively. It can be seen from Figure 3 that the Thomas correlation (Eq(5)), better fits the experimental data, with a very narrow spread of the estimates: an average ratio between calculated and experimental height (H calc /H exp ) of 1.11 with a variance of 0.03, is obtained. Also Eq(6a) gives accurate results with a slightly larger overestimation of the flame length with respect to Eq(5): average ratio H calc /H exp =1.25 and var=0.01. In Table 1 the correlation parameters for the other equations are also reported. Exp Eq (7) Eq (5) Eq (6a) Eq (6b)

Conclusions

Based on the above results, some interesting summarizing conclusions can be drawn.

As far as the burning rate of hydrocarbon liquids different than liquefied gases is concerned, when the fuelspecific data are available, the best predictions can be obtained with the Zabetakis and Burgess equation; however, when relatively large pools are taken into consideration (which often means larger than just 1-2 m), the classical correlations by Burgess and co-workers (Eqs (2) and ( 4)), can be reliably adopted, provided that accurate values of the physical properties of the involved substances are available. In the case of liquefied gases a high level of uncertainty still exists, unless more complex models are adopted which is often difficult to do in a number of application, like preliminary risk analysis.

In the case of the height of the flame generated by a circular pool fire, the Thomas equation has proved to give accurate predictions when the physical properties of the fuel are available. Quite interestingly, for a large variety of liquid mixtures, the very simple BrLtz correlation was able to accurately predict the flame height based only on the pool diameter. This is a very useful advantage when a large number of rapid calculations are required, or when a large uncertainty around the physical properties of the involved hydrocarbon mixture is present.

Figure 1 :

 1 Figure 1: Burning rates for hexane. Experimental data and models predictions.

Figure 2 :

 2 Figure 2: Burning rates for LNG. Experimental data and models predictions.

Figure 3 :

 3 Figure 3: Flame height for hexane and heptane. Experimental data and models predictions.

Figure 4 :

 4 Figure 4: Flame height for gasoline and kerosene. Experimental data and models predictions.

Table 1 :

 1 Statistical parameters for hexane and heptane pool firesIn the case of hydrocarbon mixtures, like gasoline and kerosene (see Figure4), rather different results are obtained, with the Thomas correlation showing a larger overestimation of the calculations with respect to the experimental values (H calc /H exp =1.23 average and variance 0.23), and the BrLtz correlation (Eq(7)) providing an excellent agreement with the measured data (H calc /H exp =0.99 with var=0.10). This is an interesting result, since, as highlighted in the previous section, this correlation does not require any additional input parameter, except for the pool diameter. Similar results are obtained with other mixtures (data not shown).

		Eq (5)	Eq (6a)	Eq (6b)	Eq (7)
	Average H calc /H exp	1.11	1.25	1.41	0.75
	Variance	0.03	0.01	0.03	0.02