Prasad N Samarakoon

Emmanuel Promayon

Céline Fouard

LIGHT RANDOM REGRESSION FORESTS FOR AUTOMATIC MULTI-ORGAN LOCALIZATION IN CT IMAGES

Keywords:

Classic Random Regression Forests (RRFs) used for multiorgan localization describe the random process of multivariate regression by storing the histograms of offset vectors along each bounding wall direction per leaf node. On the one hand, the RAM and storage requirements of classic RRFs may become exorbitantly high when such a RRF consists of many leaf nodes, but on the other hand, a large number of leaf nodes are required for better localization. We introduce Light Random Regression Forests (LRRFs) which eliminate the need to describe the random process by formulating the localization prediction based on the random variables that describe the random process. Consequently, LRRFs with the same localization capabilities require less RAM and storage space compared to classic RRFs. LRRF comprising 4 trees with 17 decision levels is approximately 9 times faster, takes 10 times less RAM, and uses 30 times less storage space compared to a similar classic RRF.

INTRODUCTION

In the field of medical image analysis, Multi-Organ Localization (MOL) is a preliminary step in both anatomical structure segmentation and registration. It also plays an important role in content specific image retrieval. Based on pioneering work of Breiman [START_REF] Breiman | Random forests[END_REF], Random Regression Forests (RRFs) were used in 2010 for the first time to solve the MOL problem by localizing the organs using bounding boxes [START_REF] Criminisi | Regression forests for efficient anatomy detection and localization in CT studies[END_REF]. The ingenuity of the proposal was the manner in which the MOL problem was transformed into a multivariate regression problem. It comprised the regression of a 6 Dimensional (6D) displacement vector (offset vector) from the bounding box walls of the organs to any given voxel. In 2013, the same authors proposed a modified implementation of RRFs [START_REF] Criminisi | Regression forests for efficient anatomy detection and localization in computed tomography scans[END_REF][START_REF] Criminisi | Anatomy detection and localization in 3D medical images[END_REF] that enhanced the previously achieved results by modifying the split node optimization method, the description of the random process, and the eventual usage of this description for prediction.

Since its introduction, RRFs have been successfully used as the first localization step in fully automatic segmentation of right and left kidneys [START_REF] Cuingnet | Automatic detection and segmentation of kidneys in 3D CT images using random forests[END_REF], liver [START_REF] Gauriau | A generic, robust and fully-automatic workflow for 3D CT liver segmentation[END_REF], multiple organs (liver, right kidney, left kidney, spleen, gallbladder, and stomach) [START_REF] Gauriau | Multi-organ localization combining global-to-local regression and confidence maps[END_REF], hip joint [START_REF] Chu | Facts: fully automatic ct segmentation of a hip joint[END_REF], and detection, grading and classification of coronary stenoses [START_REF] Michael Kelm | Detection, grading and classification of coronary stenoses in computed tomography angiography[END_REF] in CT volumes.

Predicting the localization of organs is based on using the offset vectors stored in the RRF leaf nodes during the training phase. The offset vectors d(v; c) are computed for each organ and for each voxel using:

d(v; c) = v -b(c) , (1)
where c ∈ organs (the set of organs to localize), b(c) the bounding box vector, and

v = (v x , v x , v y , v y , v z , v z ) made from the voxel position (v x , v y , v z ). d(v; c) is composed of 6 components (d dir c
) where dir ∈ {left, right, anterior, posterior, inferior, superior}.

Given a set of voxels arriving at a particular node and for a given organ, the current setting of RRF regresses the continuous conditional distribution of d(v; c) as a 6D multivariate Gaussian [START_REF] Criminisi | Regression forests for efficient anatomy detection and localization in CT studies[END_REF][START_REF] Criminisi | Regression forests for efficient anatomy detection and localization in computed tomography scans[END_REF][START_REF] Criminisi | Anatomy detection and localization in 3D medical images[END_REF]. Consequently, the 6D multivariate Gaussian results in 6 1D univariate Gaussians.

The authors of [START_REF] Criminisi | Regression forests for efficient anatomy detection and localization in computed tomography scans[END_REF][START_REF] Criminisi | Anatomy detection and localization in 3D medical images[END_REF] described the random process attached to the continuous conditional distribution by storing the 1D histograms of offset vectors per each wall direction per each organ. Describing the random process results in a significant storage space requirement. In addition to that, these saved 1D histograms are loaded into the RAM during the prediction phase (also called the testing phase) in order to carry out the localization of an unseen image. This again translates into a substantial RAM requirement at the prediction phase.

In this article we present Light Random Regression Forests (LRRFs) which describe the random variables inherent to the random processes that were previously described in classic RRFs. By describing the random variables, the storage and RAM requirements are drastically reduced compared to the classic RRFs.

ON GAUSSIAN DISTRIBUTION SUMMATION

The foundation of LRRF is based on the following observation on summation of 1D Gaussian distributions. A 1D Gaussian distribution (G) is described by its mean (µ) and variance (σ 2 ) in the following manner:

G ∼ N µ, σ 2 = 1 σ √ 2π e -(x-µ) 2 2σ 2 . (2)
A new distribution is made by summing many 1D Gaussian distributions together. The ultimate goal is to find the arg max of the final distribution. This is similar to adding many prediction distributions described by histograms together and finding the arg max of the final summed distribution in order to obtain the absolute bounding wall location in classic RRFs [START_REF] Criminisi | Regression forests for efficient anatomy detection and localization in computed tomography scans[END_REF][START_REF] Criminisi | Anatomy detection and localization in 3D medical images[END_REF]. Although the addition of two 1D Gaussians does not result in another 1D Gaussian, when the means of the distributions are very close, weighted means of the distributions are a good estimator of the arg max of the summation (see Fig. 1a). But when the means of the distributions are quite apart from each other, the weighted means are not a good estimator of the arg max (see Fig. 1b).

At the prediction phase of RRFs, "a good set" of leaf nodes that shows the lowest variability is selected per organ for the final localization prediction. Since each of these selected leaf node should predict approximately the correct position (hence, have approximately the same mean), we hypothesized that finding the arg max of the final prediction summation distribution can be estimated using the weighted means of each selected leaf node.

LIGHT RANDOM REGRESSION FORESTS

Similarly to RRFs, LRRFs are also an ensemble of Random Regression Trees (RRTs). Each RRT consists of split and leaf nodes. Each split node divides the incoming voxels into two separate child nodes. Leaf nodes are responsible for predicting the localization of organs.

The training phase of LRRFs is identical to the training phase of classic RRFs (see [START_REF] Criminisi | Regression forests for efficient anatomy detection and localization in computed tomography scans[END_REF][START_REF] Criminisi | Anatomy detection and localization in 3D medical images[END_REF] for a more detailed description) except for one simplification that occurs as a direct consequence of our hypothesis mention in Sect. 2. This simplification is described in Sect. 3.1 and the details of the prediction phase of LRRFs are presented in Sect. 3.2.

Training phase implementation

The split nodes are trained following the exact procedure of classic RRFs. When all the split nodes are trained, the leaf nodes contain all the accumulated voxels. Training of a given leaf node consists of summarizing the offset vectors of these accumulated voxels. As previously mentioned, RRFs regress the continuous conditional distribution of offset vectors as a 6D multivariate Gaussian which results in 6 1D univariate Gaussians along each bounding wall direction [START_REF] Criminisi | Regression forests for efficient anatomy detection and localization in CT studies[END_REF][START_REF] Criminisi | Regression forests for efficient anatomy detection and localization in computed tomography scans[END_REF][START_REF] Criminisi | Anatomy detection and localization in 3D medical images[END_REF]. The classic RRFs train the leaf node by storing the random process using histograms of the offset vectors [START_REF] Criminisi | Regression forests for efficient anatomy detection and localization in computed tomography scans[END_REF][START_REF] Criminisi | Anatomy detection and localization in 3D medical images[END_REF].

Thanks to our hypothesis, we can eliminate the storing of the histograms (i.e. the description of the random process). Hence, per organ, we only store the mean (µ dir ) and the variance (σ 2 dir ) of these 1D distributions (i.e. the random variables that define the distribution D).

D ∼ N (µ dir , σ 2 dir ) . (3)
LRRFs are "light weight" by saving a lot of storage space as they do not save the 1D histograms of each wall direction of each organ.

Prediction phase implementation

During the prediction phase, a previously unseen image is put through an already trained LRRF in order to localize the organs. Similar to classic RRFs, each voxel (v) of the image (V) is pushed through each RRT until it is accumulated at a leaf node for each RRT of the LRRF (l(v)). The set of leaf nodes that accumulates at least 75% of voxels ( L) and that displays the lowest variability per each organ is used for the localization prediction similar to [START_REF] Criminisi | Regression forests for efficient anatomy detection and localization in computed tomography scans[END_REF][START_REF] Criminisi | Anatomy detection and localization in 3D medical images[END_REF]. Then, each bounding wall distribution (D (b(c))) can be defined as :

D (b(c)) ∼ l∈ L D (b(c | l(v)) • Pr(l) , (4)
where In classic RRFs, D (b(c)) is modeled by histograms per each bounding wall direction and bc is obtained by finding the mode of the histogram (i.e. the arg max) [START_REF] Criminisi | Regression forests for efficient anatomy detection and localization in computed tomography scans[END_REF][START_REF] Criminisi | Anatomy detection and localization in 3D medical images[END_REF].

Pr(l) = |l(v)| / l(v)∈ L|l(v)|, |l (
We use our hypothesis in order to find bc by calculating the sum of the means of b (c | l(v)) weighted by Pr(l) as:

bc = l∈ L µ b • Pr(l) , (5)
where µ b = vdir -µ d . LRRFs are "light" by leading to faster computational times as no random process is mimicked by the use of histograms but a model of bc is directly built using the random variables.

MATERIALS AND EXPERIMENTS

Our dataset comprised 100 anonymized CT volumes belonging to 100 different patients that were obtained from the teaching hospital of Grenoble Alpes. These CT volumes showed a high variability in size (from 263 × 263 × 366 to 466 × 466 × 568 in mm 3 ), imaged region, noise level, with or without the injection of contrast agents, presence or absence of biomedical implants, etc. The dataset was randomly divided, similarly to [START_REF] Criminisi | Regression forests for efficient anatomy detection and localization in CT studies[END_REF], into a training set and a testing set (or prediction set) consisting of 55 and 45 volumes respectively.

The left kidney, right kidney, liver, spleen, left femur head, right femur head, left pelvis, right pelvis, and L5 vertebra were used for this study. All organs of interest of all 100 images were manually delineated using 3D bounding boxes. These manually delineated bounding boxes were used as the gold standard.

A classic RRF was trained and used for prediction as described in [START_REF] Criminisi | Regression forests for efficient anatomy detection and localization in computed tomography scans[END_REF][START_REF] Criminisi | Anatomy detection and localization in 3D medical images[END_REF]. Then, the classic RRF was transformed into a LRRF by removing the stored 1D histograms of offset vectors per each wall direction as mentioned in Sect. 3. This enabled us to compare the two methods under the exact same conditions.

We generated two results sets from classic RRFs (R Ci ) and from LRRFs (R Li ); namely:

• R C1 and R L1 with 4 RRTs and D = 12 similar to [START_REF] Criminisi | Regression forests for efficient anatomy detection and localization in computed tomography scans[END_REF][START_REF] Criminisi | Anatomy detection and localization in 3D medical images[END_REF],

• R C2 and R L2 with 4 RRTs and D ∈ [1, 2, . . . , 17], to evaluate the precision and usability of both methods.

Validation of our hypothesis: In order to evaluate the hypothesis of estimating the arg max of summation of 1D Gaussians by the weighted summation of their means as presented in Sect. 2, the following steps were carried out. First, the distribution pairs of arg max of p(b c ) and weighted means of p(b c | l) denoted respectively by D a and D µ were created for each wall direction per organ (6×9 distribution sets) using all 45 prediction images. Finally, each pair of D a and D µ were subjected to Wilcoxon-Mann-Whitney test.

Prediction precision: Bounding Wall Prediction Error (BWPE) was used as the performance evaluation metric similar to [START_REF] Criminisi | Regression forests for efficient anatomy detection and localization in CT studies[END_REF][START_REF] Criminisi | Regression forests for efficient anatomy detection and localization in computed tomography scans[END_REF]. BWPE is the absolute difference between the predicted and gold standard bounding box walls. As there are six walls per bounding box, the mean BWPE (Er) is calculated taking the mean of the six BWPEs. Er was calculated only for the organs that were fully present in the CT volumes.

Usability: The usability was evaluated using the mean time (t) and the mean amount of RAM (M ) required to do a MOL in a CT volume. The amount of disk space used to store the forest (S) was also compared.

For all statistical tests carried out, α was set to 0.01. All results were generated using a C++ implementation on a quad-core Intel R Xeon R 3GHz machine with 32GB of RAM.

RESULTS

Hypothesis verification: Wilcoxon-Mann-Whitney test failed to reject the null hypothesis of D a and D µ coming from the same underlying distribution with p-value ∈ [0.04 -0.98] for all 54 verification distribution pairs of D a and D µ . This provides strong empirical evidence that the arg max of summation of 1D Gaussians can be readily estimated by the weighted summation of their means.

Prediction precision evaluation: The mean BWPEs (Er 1 ) obtained for R C1 and R L1 per each organ are presented in Fig. 2a. The obtained Er 1 failed to reject the null hypothesis that both error measures of R C1 and R L1 for all organs originated from the same underlying distribution with p-value ∈ [0.03 -0.78]. That indicates the error is statistically the same with the two methods.

The mean BWPEs (Er 2 ) of all organs obtained were in the range of [9.85 -16.53] for R C2 and in the range of [9.99 -16.23] for R L2 . The standard deviations of R C2 and R L2 fell between [10.21 -15.17] and [9.59 -14.23] respectively. Er 2 decreased with the number of decision levels for both R C2 and R L2 .

Usability evaluation: The metrics t 2 , M 2 , and S 2 for R C2 and R L2 are presented in Fig. 2b. Both t 2 and S 2 for R C2 grow exponentially compared to the same values of R L2 . For 17 decision levels, t 2 , M 2 , and S 2 are (2.2 s, 19.5 s), (117 MB, 1147 MB), and (171 MB, 5221 MB) for R L2 and R C2 respectively. MOL using the LRRF is approximately 9 times faster, takes about 10 times less RAM, and uses about 30 times less storage space compared to the classic RRF.

DISCUSSION AND CONCLUSION

Although estimating the arg max of summation of 1D Gaussians by weighted individual means might be questionable (as the sum of 1D Gaussians is not a Gaussian distribution), the empirical results indicated that they are statistically alike.

From R C2 and R L2 , it is apparent that as the number of decision levels increase, the feasibility of classic RRFs for MOL becomes questionable. We propose LRRFs as an alternative that only describes the random variables that are inher-

Fig. 1 :

 1 Fig. 1: N 1 and N 2 are two 1D Gaussian distributions (shown in blue). N 1 + N 2 is the distribution generated by adding N 1 and N 2 together (shown in green). N 1,2 is another Gaussian distribution constructed from µ 1,2 : the weighted means (equal weights in this case) of N 1 and N 2 (shown in red). The goal is to check whether µ 1,2 is a good approximation of the arg max of N 1 + N 2 (am 1+2 ). (a) The arg max can be closely estimated by the weighted means when the means of the original distributions are closer. (b) When the means of the original distributions are far apart, taking the arg max privileges one distribution over the other.

  v)| is the number of voxels accumulated at the leaf node l, and b (c | l (v)) can be derived from (1). From (3) this translates into an addition of weighted 1D Gaussians. The final prediction of the absolute position of each bounding wall direction ( bc ) is the arg max of D (b(c)).

Fig. 2 :

 2 Fig. 2: Prediction precision and usability evaluation. (a) 1.5 Inter quartile range box plots of mean BWPEs per organ for R C1 and R L1 . (b) Evolution of t 2 , M 2 , and S 2 with number of decision levels for R C2 and R L2 .

This work was supported by the French ANR within the TECS project Robacus (ANR-11-TECS-020-01).


Fatima Zahra

Ben Moussa

Sébastien Dubois

Roland De Guio

Ivana Rasovska

Nathalie Gartiser

Rachid Benmoussa

How can TRIZ contribute to solve the problems of green logistics?

Keywords: Green supply chain management, green innovation, TRIZ

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The objective of this article is to introduce the initiated work in the framework of a thesis, on the potential contributions of creative methods of problem solving, commonly used in technical product design for green logistics problems. The studied methods are coming from a theory, TRIZ, which allows the construction of problem-solving methods, relying in particular on the dialectical and systemic.

In this first part, authors will present the TRIZ, a theory of problem solving, applied to technical systems design, and show how this theory can help to solve problems in other fields of application. Thus, the question of the contribution of TRIZ for problem solving in green logistics will be studied.

The plan of the paper will be presented at the end of this first part.

TRIZ, from technical systems design to problem solving

TRIZ Russian acronym for " Theory of Inventive Problem Solving " is a theory that was developed by G.

Alsthuller [START_REF] Altshuller | Creativity As an Exact Science[END_REF] to propose a framework for the construction of methods to identify and solve problems arising during the evolution of technical systems. He initially developed a set of tools and methods to help engineers be more creative during the formulation and problem solving phases into the 70s [START_REF] Alʹtshuller | The Innovation Algorithm: TRIZ, Systematic Innovation and Technical Creativity[END_REF]. Then, in order to develop these methods, a theoretical corpus has been developed. TRIZ is now a corpus of theory with its foundations and assumptions by which a system of methods was built. Meanwhile, new research based on TRIZ, were conducted in order to extend the TRIZ approaches to resolve any kind of problems i.e. not only those related to the evolution of technical systems [START_REF] Khomenko | Enhancing ECN's abilities to address inventive strategies using OTSM-TRIZ[END_REF]. TRIZ is based on three assumptions:  The laws of evolution of technical systems require that technical systems during their evolution follow generic trends, and that the knowledge of these laws is used to evaluate the maturity of a studied system, and thus predict evolution trend to follow and better anticipate problems to solve.  Contradictions: any system during its evolution should solve a number of contradictions that are the result of evolutionary trends, wishes, or opportunities for development of this system. The concept of contradiction is based on dialectics (in the Hegelian sense), but TRIZ adds a structured vision linking characters defining contradiction to those acting on it. This is done through two levels of formulation of the contradiction: the contradiction known as technique, which expresses the opposition character between two objectives for a system, and the physical contradiction, which defines two states of the system, satisfying each one of its conflicting objectives. (cf. Figure 1).  The specific requirements contextualize the evolution of the system, problems to solve, and therefore the solutions, depending on the environment in which the system evolves.

Figure 1: System of contradictions The extension of these laws to solve non-technical problems retains these three assumptions: the notion of contradiction as framework of problem formulation, the concept of specific condition that involves the systemic view of the studied object, and finally the concept of laws of evolution which shows that all system follows the rules in the way it evolve, these rules may or not be known according to the field of the studied object.

Based on undertaken work on the generalization of TRIZ to solve non-technical problems, some of these approaches have already been made, particularly regarding the identification and formulation of issues.

Thus, the recognized contributions of TRIZ are nowadays multiples, and concern different phases of problem solving, including:  When formulating the problem, systemic analysis methods of TRIZ provide insights for the selection of the priority problem to solve;  When analyzing the heart of the problem, contradiction model brings out the causes that prevent the resolution problem using conventional optimization techniques;  When solving the problem, the used processes lead to the construction of new models to remove the contradiction initially formulated.

Concerning the formulation phase of problems, and relying on the system of contradictions, as illustrated in Figure 1, several structural approaches have been developed in order to represent, and help, the complexity related to the evolution of a system by representing all the contradictions and the relationships between these contradictions, within a contradictions network.

Problem solving in logistics

Several approaches have traditionally been used to solve problems in logistics. Problems are addressed to solve various problems related to inventory management [START_REF] Hnaien | Gestion des stocks dans des chaînes logistiques face aux aléas des délais d'approvisionnements[END_REF], planning [START_REF] Wolosewicz | Modélisation et résolution d'un problème intégré de planification et d'ordonnancement[END_REF], [START_REF] Lemoine | Modèles Génériques et Méthodes de Résolution pour la Planification Tactique Monosite et Multi-site[END_REF], vehicle routing [START_REF] Grellier | Optimisation des tournées de véhicules dans le cadre de la logistique inverse : modélisation et résolution par des méthodes hybrides[END_REF], … In general, we can distinguish two types of methods for problem solving, some wearing more on the characterization of the problem situation, others more on finding solutions.

Concerning the characterization of the problem situation, generic approaches such as brainstorming, QQOQCP or the Ishikawa diagram are regularly used [START_REF] Crépin | Résolution de Problèmes[END_REF].

Regarding approaches more focused on finding solutions, they usually make resort to optimization algorithms, tree routing, … [START_REF] Wolosewicz | Modélisation et résolution d'un problème intégré de planification et d'ordonnancement[END_REF], [START_REF] Grellier | Optimisation des tournées de véhicules dans le cadre de la logistique inverse : modélisation et résolution par des méthodes hybrides[END_REF], [START_REF] Hnaien | Gestion des stocks dans des chaînes logistiques face aux aléas des délais d'approvisionnements[END_REF], [START_REF] Lemoine | Modèles Génériques et Méthodes de Résolution pour la Planification Tactique Monosite et Multi-site[END_REF], [START_REF] Mehrabikoushki | Partage d'information dans la chaîne logistique : 'Evaluation des impacts sur la performance d'une chaîne logistique des modes de collaboration mis en oeuvre entre les partenaires et des informations échangées[END_REF]. Thus these approaches are based on the postulate that the problem in its formulation contains elements allowing its resolution. We will see in Part 2 that this does not address all problems and only for some specific problems that TRIZ can provide complementary resolution approach to those already known.

Methods of TRIZ for logistical problems

In order to answer the question of this article, it was first necessary to specify the nature of TRIZ's contributions to problem solving in general. It is also necessary, in parallel, to identify and understand, if applicable, the specific problems related to green logistics. Thus, the state of the art which was built as part of this work was conducted with the aim to understand and better apprehend raised issues, and qualify them in terms of their resolution. Problems are they identified but not resolved, do papers bring solutions to some problems, if so, these solutions solve totally, or only partially the problem.

Plan of paper

The following sections of this article present, in a first step, the specificity of problems addressed by the TRIZ and how they differ from problems that can be solved by approaches already prevalent for the supply chain management (design). Then be proposed: first a definition of green logistics, and especially the green supply chain management, but also on the other hand, a classification of the area, with the aim of being able to identify and categorize the identified problems in the literature.

From the recognition of problems related to green supply chain management, several approaches on the contributions of TRIZ methods can be proposed:  A theoretical validation, a posteriori, which consist to analyze the identified solutions in the literature, and confront them to the principles and methods of problem solving from the TRIZ;  An applicative and operational contribution, consisting direct application of resolution principles and methods to identified problems in the literature;  A Cross-fertilization of fields related to the green supply chain management and to TRIZ methods.

TRIZ's Methods and steps to solving problems can be tested, validated and/or possibly adapted and modified to address logistical problems. Conversely, TRIZ methods can be adapted, completed, on the basis of problems and methods addressed by the logistics. Thus, the use of structuring methods already used in logistics can also be a basis for building a contradictions network representing all the problems, and their interdependence.

The nature of the contribution will be a result of the classification of green logistics domains, problems identified and solutions, but also evolution opportunities within the context of the initiated work.

Positioning of TRIZ and its problem-solving methods

In general, problem solving can be described as a 7-step process, as proposed in [START_REF] Crépin | Résolution de Problèmes[END_REF] Many problem solving approaches attach to equip and support mainly the initial steps that clarify the need, to formalize it, in order to define the expected solution. Similarly, certain approaches can sort and choose among a set of identified solutions and / or support the transition from idea, to its realization.

Step 4 remains a critical process step in the sense that there are few formal methods to support the search for solutions. This step can be described as a concept formation cognitive process as a successive amendments models.

The vision of cognitive processes as amended model is widely used in solving problems. Antonietti recalls [START_REF] Antonietti | Metacognitive knowledge about problem solving methods[END_REF] five major theoretical perspectives on which are built methods of problem solving:

-the massive production of ideas, brainstorming [START_REF] Osborn | Applied imagination[END_REF] is an usual illustration in conception; -the search for solution with combinations and arrangement of separate elements, morphological analysis where the work of De Bono [START_REF] Bono | Teaching thinking[END_REF] are based on this principle; -the use of analogies, highlighting connections between situations from different fields. Synectics Gordon [START_REF] Gordon | Synectics: The development of creative capacity[END_REF] and Altshuller's works [START_REF] Altshuller | Creativity As an Exact Science[END_REF] operate on the principle of analogy; -the vision of problem solving process as a restructuring of the encountered situation. This requires consideration of the whole problem, understand its main features and try to perceive it from different points of view. The viewpoint change is one of the principles of description of the design situation in OTSM and in functional analysis (NF X 50 100) [START_REF]NF X 50 100. Analyse Fonctionnelle -Caractéristiques fondamentales[END_REF].

-applying operators allowing transformation of the initial problem situation in a state of resolution. Decomposition of tasks, objectives, in sub-tasks and sub-objectives is an example.

All these visions share the common point of a state evolution between an initial state, which is a representation of the problem situation, and a final state, which is a representation of the solution or solution concept [START_REF]NF X 50 100. Analyse Fonctionnelle -Caractéristiques fondamentales[END_REF]. The differences are done on the level of generic states constructed and on modes of evolution between two consecutive statements. An approach that allows to distinguish the different available methods and look at the generic level and applicability of the proposed methods. Thus, well known methods such as PDCA, 8D, QRQC or the As square are a framework stipulating the different steps to follow, but then link to a toolbox to achieve effectively these steps. In this toolbox, it exists tools such as brainstorming, QQOQCP, SWOT, decision matrix, etc. Thus, it is possible to define the concept of method and tools as referring to two different systemic levels, as presented in Figure 2.

Thus, in traditional approaches, only the DMAIC which is an exception, by defining each step of the method, the tool to use and how to use it. Thus, DMAIC creates a systemic organization between the tools, and the method is more operational. Similarly, methods based on TRIZ are constructed to make explicit the link between each step. The set of tools which the TRIZ methods are used are organized systemically. Finally, since we are talking about solving problems, it is also important to characterize the problem, to see if all problems are equivalent, if they are approached in the same way by the set of resolution methods, and if these methods define the problem in the same way. In [START_REF] Rasovska | Mechanisms of model change in optimization and inventive problem solving methods[END_REF] and [START_REF] Dubois | Resolution of inventive problems: different kind of mechanisms[END_REF], authors present a characterization of the problems and especially problem solving methods, with reference to the distance traversed by these methods. Thus three areas have been defined (see figure 3):  The space of the specific problem (Specific Problem Space) is defined by the variables of the problem and the domains boundaries of these variables Di. The dimension of this space is equal to the number of variables defining the problem.



The problem space (Space Problem) is defined by variables without the limits of their domains. The dimension is always equal to the number of variables.



The space of solutions (Solution Space) is defined by all possible related variables to the problem concerned. The dimension of this space is not limited.

Figure 3: Definition of knowledge spaces Thus, these three areas have distinguished three families of methods for solving problems, optimization methods, which explore the space of the specific problem, methods for over-constrained problems that expand the space to browse through the entire problem space, and finally methods relating to problems called inventive problems (i.e. that cannot be solved by an optimization approach) that define a new model of problem representation in order to define new spaces in the solution space.

Methods based on TRIZ are specifically dedicated to this latter type of problem because they are using cognitive operators that define new variables description of the problem, in order to modify the original model description and move towards space wherein a solution can be found.

In addition, as already mentioned above, approaches from TRIZ, especially through the phases of problems description, related to a system evolution (including the construction of a network of contradictions), are an important benefit in the understanding of a problem situation.

Set of variables, their domains and constraints

Set of variables All possible variables

Problem Space (PSp) Specific Problem Space (SPS)

Space of domain solutions

Domain space extended

Universal domain

Solution Space (SSp)

Green supply chain management, what is it?

Historically, studies and research for solutions on the impacts of industrial activities have always been at the heart of the concerns of industrial world.

The growing importance of the concept of green supply chain management is mainly due to the global awareness of the environmental damage caused by pollutant releases, lower raw material resources and depletion of natural resources. However, environmental awareness is not the only driver of behavioral changes of industrial, community pressures and market demand and consumers are demanding more and more environmentally friendly products and international regulations play a large role [START_REF] Zhang | Why do firms engage in environmental management? An empirical study in China[END_REF].

However, before defining green supply chain management, we define the supply chain as [START_REF] Amrani-Zouggar | Impact des contrats d'approvisionnement sur la performance de la chaîne logistique: Modélisation et simulation[END_REF] a set of activities in networks whose execution is correlated with flows they exchange (physical, financial and information flows), to best meet the needs expressed by a set of clients.

Green supply chain management finds its roots both in environmental management and supply chain management. The definition and the application scope of green supply chain management in literature are quite large and varied. They range from "green purchasing" to taking into account the impact of pollution from the supplier to the end customer passing through production to reverse logistics which tends to find methods for management and treatment of the products end of life. Ahi et Searcy [START_REF] Ahi | A Comparative Literature Analysis of Definitions for Green and Sustainable Supply Chain Management[END_REF] have identified numerous published definitions of green supply chain management and they classified them into two groups : the one of key characteristics of sustainable development (focus on economy, focus on environment, social purpose, focus on stakeholders, focus on volunteering, focus on resilience, focus on the long term) and the other of supply chain management (focus on flows, focus on coordination, focus on stakeholders, focus on relationships, focus on value).

synthetic work proposed by [START_REF] Ahi | A Comparative Literature Analysis of Definitions for Green and Sustainable Supply Chain Management[END_REF] shows the evolution of green supply chain management concept, which was originally defined by the integration of environmental principles within the whole life cycle of a customer order [START_REF] Handfield | Green'value chain practices in the furniture industry[END_REF]. It is only from 2005, that we see the introduction of the concept of return logistics or reverse logistics [START_REF] Hervani | Performance measurement for green supply chain management[END_REF], [START_REF] Sheu | An integrated logistics operational model for green-supply chain management[END_REF]. Also, in the same year, green supply chain management has been defined differently, focusing in the definition of Zhu et al. [START_REF] Zhu | Green supply chain management in China: pressures, practices and performance[END_REF] on the realization of profits and market share objectives by reducing environmental risks and impacts while increasing the eco-efficiency of enterprises. However, the year 2007 reflects the introduction of a key factor in the evolution of green supply chain management : environmental performance [START_REF] Mida | A model for assessing the greenness effort in a product supply chain[END_REF]. Indeed, from this date, numerous research studies have focused on measuring and improving environmental performance and its impact on improving business competitiveness. In 2008, Srivastava [START_REF] Srivastava | Network design for reverse logistics☆[END_REF] exceeds the definition he proposed in 2007 [START_REF] Srivastava | Green supply-chain management: A state-of-the-art literature review[END_REF] that defines green supply chain management as integration of environmental awareness in the supply chain management as well as management of product end of life, and proposes a new definition, in which he highlights the value creation and thus introduced the principle of converting resources into usable products. In 2011, El Saadany and al. [START_REF] Saadany | Environmental performance measures for supply chains[END_REF] are interested in their definition, firstly, on the reduction of energy and virgin raw materials consumption, reduction of waste production, and secondly to increase product recovery options.

Recall here that the term « manage» a green supply chain can be understood as « to control and/or (re) design» the chain. It should be noted that the majority of definitions analyzed in [START_REF] Ahi | A Comparative Literature Analysis of Definitions for Green and Sustainable Supply Chain Management[END_REF] see « the management» as a « control element » of the chain. Few articles that discuss the design of the green supply chain as an integral part its management. Even though, the definitions presented above do not explicitly differentiate between control and design, we start initially as part of this work, by taking the following definition: «Green supply chain management is integrating environmental thinking into supply chain management, including product design, material sourcing and selection, manufacturing processes, delivery of the final product to the consumers as well as end-of-life management of the product after its useful life». [START_REF] Srivastava | Green supply-chain management: A state-of-the-art literature review[END_REF] 4. Materials and Method

Materials:

As mentioned in the introduction, the purpose of this article is to identify a scientific problem by identifying the different problems of green supply chain, and then classifying them into different categories. For this, it was necessary, in a first step, to classify the examined papers. To achieve this objective, a literature review was constructed based on research for articles published in the databases « science direct » and « IEEE » using important keywords as 'green supply chain', 'environmental supply chain', 'eco innovation', 'eco design', 'green manufacturing', 'green products', 'green transportation', 'ISO 14001' and 'reverse logistics. In total, a list of 273 articles published between 2002 and 2013 were identified.

Figure 4 shows a distribution of articles published per year over the selected years, while Figure 5 shows the distribution of the examined articles by journal. From now, we are going to detail the method that has been implemented, and which will be continued for the analysis and classification of the art's state references.

Method of Classification of papers:

To classify papers of green supply chain management of our constructed literature review, we analyzed all items so as to affect each one to classification elements proposed by Srivastava [START_REF] Srivastava | Green supply-chain management: A state-of-the-art literature review[END_REF] (figure 6).

To do this, a table including the following elements was built:  Paper's reference  Classification element Indeed, the classification proposed by Srivastava relies on the contexts of green supply chain management problems. This classification resumed in figure 3 partition articles in three major subjects (at a first level of description): green operations, green design and the importance of green supply chain management. The theme 'Green Operations' which explores the waste treatment, reverse logistics and green production / remanufacturing. To this category, two other categories join it, namely 'Green Design' which concerns risk and waste management, prevention (the safety of the product, professional health and safety, pollution) and re-source conservation. The author has also proposed another classification element, 'the importance of green supply chain management' which specifies its strategic idea and the involvement, within companies, of Top Management.

In addition to these three categories, it is interesting to note that Srivastava, delimited its research focusing only on items that address the green supply chain design problems. Thus, he consciously removed a set of papers that he considered more operational than strategic, and, according to him, it may not be important in this type of articles.

And thus, the first step of this analysis consists of allocating papers to the various categories of the first level of the classification of [START_REF] Srivastava | Green supply-chain management: A state-of-the-art literature review[END_REF] while ensuring to complete it or modify it, in case of appearance of new types of papers.

Figure 6 : Classification depending on the context of the problem in green supply chain design [START_REF] Srivastava | Green supply-chain management: A state-of-the-art literature review[END_REF].

Articles unclassified address three application domains, and thus can be grouped into three new categories, which we have called "management practices", "green procurement" and "green distribution" (Figure 7). -The importance of green supply chain management -Green design: an approach that aims to improve the ecological quality of a product, reducing its negative impacts on the environment throughout its life cycle. It is characterized by taking into account the environment during the design or product improvement stage.

-Green operations: category, which includes a set of green operations, namely:

 Green procurement or "green-sourcing": is the integration of environmental criteria in all phases of the procurement process.  Green production: is the reduction of environmental impact of production process by using less polluting materials and/or equipment to reduce polluting emissions as well as the consumption of toxic products and production waste, or by modifying the production process in order to reduce consumption of energy and raw materials, and reduce consumable wastes.  Remanufacturing: is the process of disassembly and recovery of worn components of a product in order to repair or replace them to get a product ready for reuse.  Reverse logistics and Waste treatment: is the management of logistics flows in the opposite direction: from the consumer to the source in order to recover, recycle or repair them.  Green distribution: is the set of activities that includes green transportation + green storage + green packaging + distribution network design.

-Managerial practices: are the set of strategies and practices aiming to design, coordinate, and manage the operations of the supply chain.

Classification method of current and past problems and their evolution

In this second stage of our analysis, we complete the table by including, for each reference, the following information:  Problem (s) addressed, i.e. we take here the problem as it is formulated and stated in the reference  Nature of the contribution, this one may be descriptive or prescriptive, or it may be empirical or theoretical  Comments on the solution to describe the elements of problem solving stated, as expressed in the reference  Level of resolution, three levels have been recognized as possible: no resolution, if the reference addresses a problem but it does not offer a solution, Partial, when the reference proposes an improvement way that addresses an identified problem, but does not solve it totally or reveals new problems. And finally complete resolution, when the reference identifies a problem and provides a solution which does not engender negative aspect.

This table aims to extract the green supply chain management problems treated in all papers of our literature review, and also to show the nature of the scientific contribution of each articles as well as its level of resolution, whether it is completely resolved, partially resolved or unresolved.

The objective of this step is to build the most exhaustive view possible of all the problems of green supply chain management. In a second time, these problems will be grouped into problems families to thereby provide a classification of the green supply chain management problems. It will also allow, on the basis of quantitative references by category of problems, to show the evolution of the green supply chain management problems in time, to define the typologies of emergent problems and those resolved.

Future Problems

To solve green supply chain problems, we must not only focus on the current problems, but also be able to anticipate tomorrow's problems, and it is within this context, that several prospective studies of logistics were made, and that it is in this section that we are going to identify them, to give a clear idea of the problems that will appear in the future.

Contribution of the TRIZ

After listing all the problems of green supply chain from the past to the future, we add to the table defined in section Erreur ! Source du renvoi introuvable. two columns named respectively 'potential contribution of TRIZ' and 'type of the contribution', Which consist in defining whether each problem can be solved or not with methods from the TRIZ, and if so, to identify the nature of the contribution that these methods can bring to the resolution, and which can take two aspects:

-Contribute to the understanding and formulation of the problem, a clarification step -Participate in the resolution of the problem.

Results

In this initial version of the article, only initial results are presented, and also, we described the expected results for the final version of our research work, and which will be presented as follows:

 A statistical analysis of the studied data: -Volume of publication by classification category of Figure 6.

-Volume of publication types of green supply chain management problems.

-potential contribution of TRIZ: Yes, because contribution is already made by other authors, to prove, for the contributions that seem possible, but have not yet been achieved, or not, when typology problems does not match the basic postulates of TRIZ.

-Potential contribution of the TRIZ, relating only on the clarification of the problem, and/or to the resolution of this one. As initial findings, we calculated the proportion of each types of papers category published between 2013 and 2011, in order to extend it in the next version for all papers, thus we obtain the following diagram (Figure 8): Besides, we take out from the same diagram a very small proportion for articles dealing with the importance of the green supply chain, it can be explained by two different ways: -The Research in the field is it in its initial phase? -The Research in this field does it reach its final phase?

To answer these two questions, we are going to build a diagram showing the evolution of each type of papers category, and thus to define the state of maturity of each research field. For this, we will extend the bibliographic database, to examine older articles and see if the volume of articles is in a growth or decrease phase.

The same result will be produced for green supply chain problems in the next version, and a similar diagram to paper types will be built, to show the evolution of green supply chain problems.

And finally, the last phase of our analysis is to identify problems types that can be solved or not with methods of TRIZ and their nature of contribution.

DISCUSSION -CONCLUSION

The work presented here is only the beginning of the contribution which authors plan will bring concerning the question of the contribution of TRIZ methods to problems related to the green logistics. The continuation of the study will aim to extend the studied bibliographic database to allow a quantitative analysis which the interest is to be able to identify emerging problems of those being addressed.

We have already seen that there is a difference regarding the potential contribution of resolution methods between a classification that identifies fields of contribution (as proposed by Srivastava) and a classification which focuses on the nature of the problems, regardless the domain in which the problem appears. However, already the first elements show clearly that we can distinguish three types of contributions:  Green product design and/or process, here contributions have already been made and therefore the contribution of TRIZ methods are proved.  Green supply chain design, where different problems can occur, and we have no doubt here in the interest of methods, at least for the structuring and formulation of problems, remains only to see the contribution to the resolution phases.  The green control of supply chain, still a field of investigation. Even how methods already used in logistics and those out coming from TRIZ can enrich and even the nature of problems to be solved and if methods from TRIZ can help.

Figure 2 :

 2 Figure 2: Differentiation methodtools of resolution

Figure 4 :Figure 5 :

 45 Figure 4: distribution of the examined articles per year *Note: Data for 2013 include only published articles until August

Figure 7 :

 7 Figure 7: completed classification of Srivastava

Figure 8 :

 8 Figure 8: Proportion of problem categories of green supply chain management This diagram shows that green operations occupy the interest of researchers in recent years. This reflects the high awareness of industrialists and consumers for green and environmentally friendly consumption.