Prof. Dr. habil Dmitry Ivanov
email: divanov@hwr-berlin.de

Alexander Pavlov

Alexandre Dolgui
email: alexandre.dolgui@mines-nantes.fr

Boris Sokolov

Hybrid fuzzy-probabilistic

Keywords: supply chain engineering, risk management, resilience, structure dynamics, ripple effect, graph theory, fuzzy theory

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Introduction

The supply chain engineering under uncertainties is a major issue in production systems and logistics [START_REF] Dolgui | Supply chain engineering: useful methods and techniques[END_REF]. One of most challenging problems deals with an increase of supply chain resilience, especially facing ripple effect.

The ripple effect, also known as domino effect or snowball effect [START_REF] Wierczek | The impact of supply chain integration on the "snowball effect" in the transmission of disruptions: An empirical evaluation of the model[END_REF]) describes the disruption propagation in the supply chain (SC), the impact of a disruption on SC performance and the disruption-based scope of changes in the SC design (SCD) structures [START_REF] Liberatore | Hedging against disruptions with ripple effects in location analysis[END_REF], Ivanov et al. 2014a,b, Sokolov et al. 2016). Managing the ripple effect is closely related to designing resilient SCs.

SC resilience became one of the key investigation categories over the last decade [START_REF] Gunasekaran | Supply chain resilience: role of complexities and strategies[END_REF]. SC resilience is understood in literature as an ability to maintain, execute and recover (adapt) the planned process states along with achievement of the planned (or adapted, but yet still acceptable) performance [START_REF] Sheffi | A supply chain view of the resilient enterprise[END_REF], Ivanov and Sokolov 2013, Kamalahmadi and Mellat-Parast 2016a,b). Recent studies underline the crucial role of resilience assessment taking into account disruption risks (e.g., Ivanov et al. 2013[START_REF] Munoz | On the quantification of operational supply chain resilience[END_REF][START_REF] Das | Risk readiness and resiliency planning for a supply chain[END_REF], Ivanov et al. 2016).

The existing studies have mostly focused on the impact of disruptions on SC resilience using deterministic approaches and the reliability theory without considering structure reconfiguration [START_REF] Wagner | Assessing the vulnerability of supply chains using graph theory[END_REF][START_REF] Soni | Measuring supply chain resilience using a deterministic modeling approach[END_REF][START_REF] Kim | Supply network disruption and resilience: A network structural perspective[END_REF]. Recently, the necessity of inclusion the reconfiguration into SC resilience analysis was manifested by [START_REF] Simchi-Levi | From superstorms to factory fires: Managing unpredictable supply chain disruptions[END_REF], [START_REF] Xu | Predicted supply chain resilience based on structural evolution against random supply disruptions[END_REF], [START_REF] Snyder | OR/MS Models for Supply Chain Disruptions: A Review[END_REF]. However, to the best of our knowledge there is no published graph-theoretical research on SCD resilience analysis with both disruptions and recovery considerations. In addition, the existing studies (e.g., [START_REF] Kim | Supply network disruption and resilience: A network structural perspective[END_REF]) consider monotonous SC structures (i.e., the failures of structure elements lead to the reliability decrease and recovery of structure elements lead to SC reliability increase). In reality, SC structures can be nonmonotonous, i.e., the factors such as competitor behavior can also influence the SC resilience (e.g., a failure of one competitor can lead to reliability increase).

In this study, the following problem is considered. The object of investigation is a multi-stage SC different elements of which can be disrupted. The problem is to identify the impact of disruptions at different SC locations on SC resilience and to identify the critical SC nodes and arcs the disruptions of which result in SC resilience loss.

The goal of this study is to develop a method for quantitative analysis of SCD resilience with disruption propagation and recovery considerations for both monotonous and non-monotonous structures. We are also interested in developing an SCD resilience index that can be used by SC managers to compare different SCDs regarding the resilience.

The remainder of this paper is organized as follows. Section 2 provides a state-of-the-art overview. In Section 3, the research methodology is presented. Section 4 describes the fuzzyprobabilistic approach and the underlying concept of graph genome. In Section 5, the modeling procedure for the SC structure dynamics with reconfiguration is described. In Section 6, the SCD resilience index is described. Section 7 is devoted to the experimental part. In Section 8, managerial insights are presented. The paper concludes by summarizing the most important insights from the research and delineating future research avenues.

Literature review

The ability to maintain and recover (adapt) the planned execution along with achievement of the planned (or adapted, but yet still acceptable) performance is related by most authors to SC resilience [START_REF] Bakshi | Co-opetition and investment for supply-chain resilience[END_REF], Ivanov and Sokolov 2013[START_REF] Xu | Predicted supply chain resilience based on structural evolution against random supply disruptions[END_REF][START_REF] Ambulkar | Firm's resilience to supply chain disruptions: Scale development and empirical examination[END_REF]. It can be observed in the existing studies that two groups of problem statements for SC resilience analysis are generally considered (see Fig. We limit the literature analysis to graph-theoretical approaches taking into account the focus of this research. Stochastic Petri nets have been applied to analyse disruption propagation through the SC and to evaluate the performance impact of the disruptions [START_REF] Wu | Methodology for supply chain disruption analysis[END_REF]. [START_REF] Wang | Evaluation and analysis of logistic network resilience with application to aircraft servicing[END_REF] and [START_REF] Soni | Measuring supply chain resilience using a deterministic modeling approach[END_REF] develop resilience metrics with the use of graph theory. [START_REF] Wagner | Assessing the vulnerability of supply chains using graph theory[END_REF] propose a method of quantifying risk using the permanent of an adjacency matrix based on graph theory. [START_REF] Nair | Supply network topology and robustness against disruptions -An investigation using a multi-agent model[END_REF] analyse the correlation between disruption and structural features of network. [START_REF] Hsu | Reliability evaluation and adjustment of supply chain network design with demand fluctuations[END_REF] develop a method to evaluate reliability of the SC performance under the demand fluctuations. [START_REF] Schoenlein | Measurement and optimization of robust stability of multiclass queuing networks: Applications in dynamic supply chains[END_REF] define robustness as the ability of a multiclass queuing network to remain stable, if the expected values of the inter-arrival and service time distributions are subject to uncertain shifts. Based on the fluid network analysis they present a measure to quantify the robustness called the stability radius. Ivanov and Sokolov (2013) describe SC stability, robustness, and resilience as an integrated control problems within the framework of automatic control theory. [START_REF] Zobel | Characterizing multi-event disaster resilience[END_REF] quantify resilience of supply chain for multiple disruptive events. [START_REF] Xu | Predicted supply chain resilience based on structural evolution against random supply disruptions[END_REF] develop a quantitative model for analysis of predicted SC resilience based on structural evolution against random supply disruptions. The study by [START_REF] Lin | Network reliability with deteriorating product and production capacity through a multi-state delivery network[END_REF] concentrates on the reliability assessment for a multi-state SC with multiple suppliers as the probability to satisfy the market demand within the budget and production capacity limitations.

They develop an algorithm in terms of minimal paths to evaluate the network reliability along with a numerical example regarding auto glass. Garvey et al. (2015) build upon minimal paths analysis and suggest using Bayesian network to analyse risk propagation in the SC. This is an interesting research avenue. Bayesian networks have also been applied to domino effect analysis in chemical industry infrastructures [START_REF] Khakzad | Application of dynamic Bayesian network to risk analysis of domino effects in chemical infrastructures[END_REF].

Another important research stream has been the understanding that different kinds of SCD structures with similar number of nodes and arcs may significantly differ regarding resilience. [START_REF] Bode | Structural drivers of upstream supply chain complexity and the frequency of supply chain disruptions[END_REF] analyse empirically structural drivers of upstream SC complexity and the frequency of SC disruptions. They find out dependencies between SC structural complexity and disruption occurrence. [START_REF] Kim | Supply network disruption and resilience: A network structural perspective[END_REF] apply graph theory to analyse the impact of SC structural characteristics on resilience. This study reveals correlations between network structure and disruption impacts. [START_REF] Brandon-Jones | The impact of supply base complexity on disruptions and performance: the moderating effects of slack and visibility[END_REF] find out the impacts of supply base complexity on disruptions and performance. [START_REF] Sokolov | Structural analysis of the ripple effect in the supply chain[END_REF] quantify ripple effect in the SC with the help of selected indicators from graph theory and develop a hybrid static-model model for performance impact assessment of disruption propagation in a distribution network. [START_REF] Han | Evaluation mechanism for structural robustness of supply chain considering disruption propagation[END_REF] assess the SC structural robustness considering disruption propagation in a connected graph. They perform quantitative assessment of the structural robustness on random networks compared with the probability of network disruption due to the random risk. [START_REF] Tang | Complex interdependent supply chain networks: Cascading failure and robustness[END_REF] develop a time-varied cascading failure model and analyse the ripple effect as failed loads propagation in the SC. They present the SC as interdependent structure of an undirected cyber network and a directed physical network, two layers that constitute an SC. The authors develop a robustness measure and analyse the SC collapse situations. Some critical observations can be derived from this literature review. Typical assumptions that an increase in SC resilience depends on the number of nodes and arcs in the SC (i.e., back-up suppliers or alternative transportation routes) frequently lead to the solutions where SC resilience can be increased only if SC efficiency decreases. In reality, SC structures are different and the SCDs with the same efficiency can perform completely differently in regard to resilience. This dependence of the SC resilience on the SCD structure has not been studied in literature extensively so far. Furthermore, the performance analysis with the use of individual supplier failure probabilities dominates the research domain. At the same time, another important question concerning the disruption propagation and SCD survivability is still at the early stage of investigations. In the scope of the research described in this article is the analysis of what SC elements will survive (i.e., be in operation) after a disruption and under which conditions (i.e., joint failures in a group of suppliers or a failure at a critical supplier that interrupts the SC operation fully) the SC can survive or it will lose its survivability. Finally, the issues of SC reconfiguration have got only limited attention in recent literature and they will be included into our study.

Research methodology

The research concept of this study is based on the following four stages:  Theoretical analysis of the hybrid fuzzy-probabilistic approach with the application to SCD resilience analysis  Modelling the SC structure dynamics with reconfiguration  Development of an SCD structure resilience index (SRI)  Validation of the SCD SRI according to the referenced SCD structures from literature.

In Section 4, theoretical analysis of the hybrid fuzzy-probabilistic approach with the application to SCD resilience analysis will be elaborated. In this part, we define the underline concept of this study, the graph genome. It will be shown how to develop a SCD genome and how to compute its components on an example. Subsequently, the formulas for computing the structure failure probability for both monotonous and non-monotonous structures taking into account both homogenous and heterogeneous failure probabilities and possible failures will be presented. Theoretical genome properties will be analyzed and illustrated on an example.

Section 5 deals with modelling the SC structure dynamics with reconfiguration. In this section, the central issue is the description of scenarios for modeling SCD failure and reconfiguration. In addition, in this section two underlying optimization problems for computing the failure probabilities in the cases with and without reconfiguration will be defined.

Section 6 integrates the materials of Sections 4 and 5 and develops SCD structure resilience index. First, on the basis of the Section 5, an example of SCD reconfiguration path is presented.

Second, the computation of the SRI with the help of the genome concept from Section 4 is presented.

Section 7 is devoted to the validation of the SCD resilience index according to the referenced SCD structures from literature. First, we present the SCD structures from study by [START_REF] Kim | Supply network disruption and resilience: A network structural perspective[END_REF] in terms of fuzzy-probabilistic approach. Second, we compute the resilience index values for three cases: worst-case and no reconfiguration, average case and no reconfiguration and average case with reconfiguration. The results are compared with each other and analyzed.

Hybrid fuzzy-probabilistic approach

The research methodology is based on a hybrid fuzzy-probabilistic approach to network reliability analysis.

For both monotonous and non-monotonous structures, we consider genome that can be represented as a vector [START_REF] Kopytov | New methods of calculating the Genome of structure and the failure criticality of the complex objects' elements[END_REF]). The genome is composed of the integer coefficients of the structure failure polynomial (Eq. 1) [START_REF] Aggarwal | A New Method for System Reliability Evaluation[END_REF][START_REF] Ryabinin | Reliability of Engineering Systems[END_REF], Colbourn 1978):

0 1 2 ( , , ,..., ) n       (
1 2 1 1 1 1 2 1 1 ( , ) ( , ,..., ) { } ( ) ( ) ( ) ... ( 1) ( ... ) ( 1) t m i m n t i i j t i i j C m m t i j k m q i j k t i q V R t T Q Q Q P R P R P R R P R R R P R R R Q                              , (1)
where

( 1 2 ... n Q Q Q Q     ); 2 0 1 2 ( ) ... n n T Q Q Q Q          . 1 2 ... n Q Q Q Q    
In the case of a monotonous structure, m is minimum number of failure edge-cuts, t m C is the number of combinations from m for t, ( , ) i V R t is the set of SC elements from {1, 2,..., } n , where n is the total number of SC elements in the SCD structure graph G= (V, E), i.e., the number of vertices and edges in the graph that can be disrupted plus one element. i R is minimum failure edge-cut (i.e., the edge-cut that does not contain any further failure edge-cuts); in the case of an element failure in the edge-cut, the net-work becomes divided into two non-linked parts, with the source node in the first part, and the target node in the second part, ,   are the logical operators "and" and "or" respectively, q Q is the failure probability of a network element.

The objective of the genome method application to the SCD resilience analysis is to include the structural properties of the SCD into resilience assessment. The next specific feature of the genome method is the usage of minimum structure failure edge-cuts that allows identifying groups of critical suppliers or a critical supplier. This means that disruptions at the minimum structure failure edge-cut separate the SC into two non-connected parts and operations are interrupted. If the minimum edge-cut comprises only one element, this element is called "bridge" [START_REF] Deistel | Graph Theory[END_REF]).

In Fig. 2, the monotonous network structure of an SC is depicted.

Fig. 2. Monotonous network structure of an SC

The simplified SC in Fig. 2 comprises five nodes (nodes #1 and #5 are source and target respectively) and ten arcs. For the situation, where disruption may happen only on the arcs, the genome of this SCD includes 11 elements. If disruptions would also be considered for nodes, the genome would include 16 elements. For simplification of the explanation we assume that the nodes are reliable and the failures are subject to the arcs only. In this case, for the given SCD there are five minimum failure edge-cuts as follows (Eq. 2):

{ , }, { , , , , }, { , , , , , }, { , , , , }, { , , , , , } R Q Q R Q Q Q Q Q R Q Q Q Q Q Q R Q Q Q Q Q R Q Q Q Q Q Q      (2)
The failure polynomial for this SCD can be represented as follows (Eq.3):

1 2 10 1 10 2 3 7 9 10 3 4 6 7 9 10 5 6 8 9 10 2 4 5 8 9 10 1 2 3 7 9 10 1 3 4 6 7 9 10 1 5 6 8 9 10 1 2 4 5 8 9 10 2 3 4 6 7 9 10 2 3 5 6 7 8 9 10 2 3 4 5 7 8 9 10 3 4 5 6 7 8 ( , ,..., )

T Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q             
9 10 2 4 5 6 8 9 10 1 2 3 4 6 7 9 10 1 2 3 5 6 7 8 9 10 1 2 3 4 5 7 8 9 10 1 3 4 5 6 7 8 9 10 1 2 4 5 6 8 9 10 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9

10 Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q           (3)
or in a more compact form as Eq. ( 4):

2 5 7 8 9 10 ( ) 2 4 5 2 . T Q Q Q Q Q Q Q       (4)
In the example in Fig. 2, we have m=5, number of combinations from m for t (t=1,2,3,4,5; i.e., the combination number for 1,2,3,4 and 5 edge-cuts from five minimum edge-cuts) equals 5,10,10,5,1 respectively, n=10 edges. The first five members in 1 2 10 ( , ,..., ) T Q Q Q (Eq. 3) corre- spond to t=1 (i.e., the edge-cuts from five minimum edge-cuts). Next nine members in Eq. ( 3) with "-" are related to t=2 (there should be ten members but one member is cancelled by "+" in t=3). Next seven members with "-" are related to t=3 (there should be ten members but one member is cancelled by "-" in t=2 and two members are cancelled by "-" in t=4 ). Finally, the last two members with "-" relate to t=4 (there should be five members but two members are cancelled by "+" in t=3 and one member is cancelled by "+" in t=5 ). The genome of this SCD is (0,0,1,0,0, 2,0, 4, 1,5, 2)

   
 .The interpretation of this genome is as follows:

-The SCD contains 10 transportation edges (vector dimensionality is 11=10+1)

-The components of the genome change their operators ("+" and "-") three times. This means we have three production sites in the SC (vertices #2, #3 and #4). Node #1 is a supplier, and node #5 is a customer -The last genome component is "-2". This means, that the graph has an edge to ensure the direct and return deliveries (i.e., the edge Q 4 is a "bridge" that makes it possible to deliver products from node #3 to # 4 and return the trucks from #4) -The first non-zero genome component "1" is the third one. This means that the SCD has one edge combination that contains two edges the failure of which would result in full SC destruction (i.e., the edges 1 10 { , } Q Q ).

-The second non-zero component is "2" and it is the sixth genome component. This means that the SCD has two edge combinations each of which contains five edges the failure of which would result in full SC destruction (i.e.,

2 3 7 9 10 { , , , , } Q Q Q Q Q and 5 6 8 9 10 { , , , , } Q Q Q Q Q ).
Insight 1. The usage of minimum structure failure edge-cuts allows identifying groups of critical suppliers or a critical supplier whose failure separates the network into two disconnected parts and interrupts the SC operation. For SC elements at the minimum structure failure edge-cuts, back-up sourcing strategies need to be implemented.

Furthermore, the genome encompasses the following properties of the monotonous and nonmonotonous structures:

 if 0 0   and 1 2 ... 1 n        then the failure polynomial 2 0 1 2 ( ) ... n n T Q Q Q Q          describes the monotonous structure;  if 0 1
  and 1 2 ...   and 1 2 ...

0 n        ,
1 n       
, then the structure is non-monotonous and for the failure polynomial is true (0) 1 T  .

In addition, the following topological properties of the monotonous structures are contained in the genome:

 the power of the lower polynomial member is equal to the minimal power among minimum structure failure edge-cuts (i.e., the number of the first non-zero genome component with 0 0,

l i è i l       )
; the coefficient of the lower polynomial member is always positive and equals the number of the minimal-power minimum failure edge-cuts (i.e., 0 0,

l i è i l       );
 the power of the highest polynomial member is equal to the number of the network elements.

The genome can be used for computing the structure failure probability for both monotonous and non-monotonous structures taking into account both homogenous (F hom ) and heterogeneous (F het ) failure probabilities and possible (F possib ) failures according to Eq. ( 5):

1 1 1 ( ) (1, , ,..., ) 2 3 1 T hom F n      , 2 1 1 1 ( ) (1, , ,..., ) 2 2 2 T het n F     , 2 [0,1] ( ) sup min{ (1, , ,..., ) , ( )} n T possib F g           .
(5)

In the case of homogenous failure probabilities, the network failure probability for the polyno-

mial 2 0 1 2 ( ) ... n n T Q Q Q Q          is in the interval [0,1].
The closer the failure function to the line ( ) 1 T Q  , the lower is the network reliability. This property allows using Eq. ( 6) for computing the indicator of the network structure failure:

1 1 2 0 1 2 0 1 2 0 0 1 1 1 ( ) ( ) ( ... ) ( , , ,..., ) (1, , ,..., ) 2 3 1 n T hom n n F T Q dQ Q Q Q dQ n                     . ( 6
)
For heterogeneous failure probabilities, Eq. ( 7) is used:

1 1 1 1 2 1 2 0 1 2 2 0 0 0 1 1 1 ( ) ... ( , ,..., ) ... ( , , ,..., ) (1, , ,..., ) 2 2 2 T het n n n n F T Q Q Q dQ dQ dQ            . ( 7
)
However as pointed out in recent literature (Lim et al. 2014[START_REF] Simchi-Levi | From superstorms to factory fires: Managing unpredictable supply chain disruptions[END_REF], the usage of failure probabilities in SCD decisions is quite restrictive. Such estimations can be problematic if not enough statistical data is available to fairly estimate the failure probabilities. For this reason, the alternative approach can be the usage of the fuzzy-method based on the fuzzy possibility space [START_REF] Singer | A fuzzy set approach to fault tree and reliability analysis[END_REF]).

The fuzzy possibility space can be described as ( , ( ), ) X X P



, where X is a set of elementary events, ( ) X  is σ-algebra of subsets in the space X , called the events, and the functions : ( ) [0,1] P X   , called the possibility measure (i.e., ( ) P A is the possibility of the event ( ) A X

 

). The event possibility is defined by the possibility distribution function : [0,1] g X  as ( ) sup ( )

x A P A g x  
. In the possibility measure space, the fuzzy element ( )

    ( : [0,1] X   )
is defined. Its membership function can be described as the possibility distribution function

  ( ) ( ) ( ), x P x g x x X      .
Further assume that the SC network structure comprises ( ), 1,2,...,

i i i n      fuzzy elements.
Using the operators for fuzzy structures ( 1

2 1 2 1 2            , 1 2 1 2        ),
        ) can be described as 2 0 1 2 ( ) ... n n T              , where 0 1 2 ( , , ,..., ) n       is the structure genome.
Similar to the probabilistic approach, we consider the computational procedure for the indicator ). These functions encompass the following properties:

1.

[0,1], ( ) ( ) f L X f L X          . 2. ( ), :[0,1] [0,1] f L X v    
are the involutions of the monotonously decreasing function

(0) 1, (1) 0 v v   ) ( ( )) ( ) v f v f x L X     (Pyt'ev 2002). 3. ( ), 1,2,.... i f L X i     1 1 ( ) sup ( ) ( ), ( ) inf ( ) ( ) i i i i i i i i f x f x L X f x f x L X           .
Integral : ( ) p L X L  for the possibility measure ( ) sup ( )

x A P A g x  
at the distribution g is defined as:

( ) sup min{ ( ), ( )} x X p f f x g x  
. If the possibility measure is defined by the distribution

( ) g L X 
, then the following is true (Eq. 8):

[0,1] [0,1] [0,1] ( ) 0 ( ) ( ) sup min{ , ({ ( ) })} sup min{ ,sup{ ( ) ( ) }}
sup sup min{ , ( )} sup sup min{ , ( )} sup min{ ( ), ( )} ( ).

x

X x X x X f x x X f x s f P x f x g x f x g x g x f x g x p f                     (8),
where ( ) s f is classical Sugeno fuzzy integral [START_REF] Sugeno | Theory of fuzzy integrals and its applications[END_REF].

The polynomial of the structure failure possibility This property allows using as an integral indicator of the structure failure possibility the integral for the possibility measure (Eq. 9) and provides evidence for structure failure probabilities defined in Eq. ( 5):

2 0 1 2 ( ) ... n n T              is
2 [0,1] [0,1] ( ) sup min{ ( ), ( )} sup min{ (1, , ,..., ) , ( )} n T possib F T g g                . ( 9
)

Modelling the SC structure dynamics with reconfiguration

As a consequence of a disruption, a SCD can have the form of different states S. A state represents the SCD (i.e., the graph G=(V,E)) subject to non-disrupted and disrupted elements. Due to the fact that the SC elements can recover after a failure it should be noted that the initial state S 0 denotes the state where no SC element is failing. Subsequently, due to disruptions and recovery actions, the SCD graph can transit through different states. An example for five state levels is given in Fig. 3.

Fig. 3. Levels of the SC structure dynamics

The first degradation level corresponds to those states with the failure in one element; the second degradation level corresponds to the states with the failure in two elements, etc. The recovery actions move the SCD back from the second level to the first level, and so on. The sequence of the transitions through the structural states is called SC structure dynamics control [START_REF] Ivanov | A multi-structural framework for adaptive supply chain planning and operations with structure dynamics considerations[END_REF][START_REF] Ivanov | Structure dynamics control approach to supply chain planning and adaptation[END_REF]. The sequence of events for the structure dynamics is called the structure dynamics scenario.

Scenarios belong to crucial issues in the SCD resilience analysis. In general, we do not know exactly which elements will fail. However, the structure failure polynomial (Eqs 1 and 2) contains the graph structure. For example, in the case of Q 1 failure, we have an indirect failure of andQ 9 . In other words, the SCD will transit in a state where

Q 2 ,Q 3 ,Q 4 ,Q 5 ,Q 6 ,Q 7 ,Q 8 ,
Q 1, Q 2 ,Q 3 ,Q 4 ,Q 5 ,Q 6 ,Q 7 ,Q 8
, and Q 9 are missing.

In addition, failures in different elements may have different impact on the SCD resilience. That is why we suggest analyzing homogenous, heterogeneous and possibilistic failures, respectively.

Finally, recovery actions may influence the resilience. That is why we suggest considering three scenarios, i.e., worst case (failure in the most critical elements and no recovery measures), average case and no reconfiguration, and average case with reconfiguration (see Sect. 6).

Since genome represents the SCD, each structural state S  can be described by a genome   .

Therefore, the total resilience or total failure of a path in the SC structure dynamics can be com-puted using Eqs ( 6), ( 7) and ( 9). With the help of integrating these indicators, SCD resilience can be quantified for homogenous, heterogeneous and possibilistic failures and recovery actions.

Consider the SC reconfiguration processes. Denote as

j Q   
 the transition from one struc- tural state S (described as a genome  ) to another structural state S  (described as a genome   )

as a consequence of a failure/recovery in a graph element ĵ Q Q  . Denote as ( ) X  the set of all structural states subject to  . Then, a reconfiguration scenario in the degradation or recovery process subject to an initial state 0 S and the state f S can be described as the following transition chain (Eq. 10):

3 1 1 2 0 1 2 1 ... , j j j j j N N N N Q Q Q Q Q                      (10) where 0 0 , N f         , 1 2 { , ,..., } N j j j Q Q Q Q  The structural
Q Q Q Q                    (11)
To solve the problem (11), a hybrid branch-and-bound/evolutionary algorithm has been developed as follows.

Step 1. At each k -iteration, a random sequence is built

0 1 2 1 ( ) ( ) ( ) ( ) , , , ..., , N N k k k k                   (where 0 0 , N f        
) that corresponds to a structure reconfiguration path.

Step 2. For the structure reconfiguration path, the indicator Step 3. The value of

1 ( ) ( ) 1 ( ) ( ) j N k k failure failure j F F         is com-
1 ( ) ( ) 1 ( ) ( ) j N k k failure failure j F F        
for the structure reconfiguration path is compared with the F-value at k-1 iteration.

Step 4. The path with highest F-value is used in the following iterations.

The problem of pessimistic structure dynamics path (i.e., the case where no recovery actions are included) can be described as the following optimization problem (12):

0 1 1 2 0 0 ( ) , , ( ) , { , ,..., } . j N j j N N failure j f j j j maximize F subject to X Q Q Q Q                    (
( ) ( ) 1 0 0 ( ) ( ) 2 j j k k N failure failure k j F F S          
describes the total structural resilience for a given reconfiguration path by changes in the structure failure values during the reconfiguration on the path ( ) k   . In this case, the relation (Eq. 13) describes the SCD resilience during the struc- tural reconfiguration on the path

( ) k   0 k k k S J S  (13)
( ) ( ) ( ) ( ) , , , ..., , N N k k k k                   (where 0 0 , N f        
) is built according to a SCD structure reconfiguration path. For the path, the value of

0 k k k S J S 
is computed. Finally, the average of all experiments is computed as shown in Eq. ( 14).

0 1 1 M k k J J M    (14)
The SCD structure resilience SU J belongs to the interval [ min J , max J ] and the expected SC resilience is 0 J . The values of SU J can be described as a fuzzy triple ( , , a   ), where

0 a J  , 0 min J J    , max 0 J J    .
Considering three cases (monotonous structure, non-monotonous structure and structure with possible failures), three fuzzy triples are computed ( , ,

î î î a   ), ( , , n n n a   ), ( , , a      ).
Then the integral SCD structure resilience SU J  is computed as Eq. ( 15):

( , , ) ( , , ) ( , , ) ( , , ) 3

î î î n n n SU a a a J a                    (15)
Finally, the center-of-gravity method is used to avoid the fuzziness in the final solution (Eq. 16):

( ) 3 SU E J a          . ( 16
)

Experimental results

For analysis, we consider four SC network structures from the study by [START_REF] Kim | Supply network disruption and resilience: A network structural perspective[END_REF] (Fig. 5). According to the model and algorithm in Sect. 5, the modelling of the worst case (failure in the most critical elements and no recovery measures), average case and no reconfiguration, and average case with reconfiguration subject to both nodes and arcs failures has been performed. Similar to [START_REF] Kim | Supply network disruption and resilience: A network structural perspective[END_REF], we assumed that the node #1 (Target) and node #12 (Source) are not perturbed. The computed values of SCD resilience are presented in Table 1. In comparing the results of these methods, it can be observed that all methods suggest the same order of the resilient SCD: BCDA. At the same time, the difference in the absolute resilience values can be observed in comparing the results of [START_REF] Kim | Supply network disruption and resilience: A network structural perspective[END_REF], our results without reconfiguration and our results with reconfiguration. Besides, in the study by [START_REF] Kim | Supply network disruption and resilience: A network structural perspective[END_REF], the SCD #B has a significantly higher resilience as compared to three other structures. In our analysis, the resilience of B and C is close to each other. The explanation of these effects can be seen in the inclusion of the reconfiguration and fuzzy-probabilistic analysis into resilience index computation.

Insight 2. SC resilience depend both on network structure characteristics and recovery policies.

Fair SC resilience estimation needs to include both proactive and reactive policies in order to compare different SCDs regarding the resilience. For SC elements at the minimum structure failure edge-cuts, proactive strategies need to be integrated with recovery policies.

In addition, the inclusion of the minimum edge-cut failures in the genome (instead of individual failures in the reliability analysis) is seen as the explanation of the effects in computational results. This is the benefit and novelty of the proposed approach. It can also be observed that except for structure (b), the difference of the values of [START_REF] Kim | Supply network disruption and resilience: A network structural perspective[END_REF] and our results in the case with reconfiguration is approximately 0.2. The reason for this phenomenon is, on one hand, the inclusion of the pessimistic scenario (i.e., worst-case and no recovery actions) into our resilience index. Additionally, similarities in failures (to ensure the comparativeness of the results) lead to such a correlation. Finally, higher resilience values in our approach can be observed. If comparing the results of our approach with and without reconfiguration, it can be noted that the proposed method to compute structure resilience index works in both cases. Its values also reflect clearly that the inclusion of reconfiguration increases the SCD resilience. This insight requires more analysis efforts subject to different recovery policies but we can assume the effect of the SC structure reconfiguration as resilience increase driver.

Managerial Insights

Disruption risks may result into ripple effect and structure dynamics in the SC. It is to notice that the scope of the rippling and its performance impact depend both on robustness reserves (e.g., redundancies like inventory or capacity buffers) and speed and scale of recovery actions (Knemeyer et al. 2009[START_REF] Hu | Managing risk of supply disruptions: Incentives for capacity restoration[END_REF], Ivanov and Sokolov 2013[START_REF] Kim | Guilt by association: Strategic failure prevention and recovery capacity investments[END_REF][START_REF] Pettit | Ensuring supply chain resilience: Development and implementation of an assessment tool[END_REF]. In many practical settings, companies need analysis tools to estimate both the SC efficiency and SC resilience. For SC resilience, the impacts of recovery actions subject to different disruptions and performance indicators need to be estimated.

The results of this study contribute to support decisions in these practical problems. The developed model can help the SC risk managers to identify whether the existing SCD is resilient for different disruption scenarios. The model also considers mitigation strategies (i.e., reconfiguration) that can be used by SC risk managers and translated into the SCD changes.

With the use of the developed approach, SC managers can compare different possible SCDs regarding their resilience using the proposed SCD resilience index. Since the calculation of the resilience index includes the recovery actions, the developed model can help to identify opportunities to reduce disruption and recovery costs by SC re-design. If the experts can forecast or at least assume possible failures and recovery actions, it becomes possible to create reconfiguration scenarios (see. Fig. 4) and compute the SCD resilience index subject to optimistic, pessimistic or possibility failure scenarios.

The original feature of this study is that it allows analyzing the groups of critical suppliers or a critical supplier whose failure interrupts the SC operation fully. The identification of such critical suppliers with the help of genome analysis and minimum edge-cuts usage may allow redesigning the SC in order to increase the SCD resilience without efficiency decrease. In addition, the identification of nodes and arcs in the minimum edge-cuts in the SC the failure of which is critical for SCD survivability may help to develop more specific risk mitigation policies such as dual sourcing or continuous risk monitoring.

With the help of the structure dynamics control, the model analyses effective ways to recover and re-allocate resources and flows after the disruption. The model can be used by SC risk specialists to adjust mitigation and recovery policies with regard to critical SCD elements. Finally, the usage of the fuzzy-probabilistic approach extends possible application of the developed method regarding the data availability for analysis.

Conclusions

In recent literature, SCD models have been extensively considered in the light of severe disruptions. SC resilience became one of the key investigation categories. The existing graphtheoretical studies have mostly focused on the impact of disruptions on SCD resilience using the reliability theory without considering structure reconfiguration. In this paper, we take another perspective and extend the existing models to SCD resilience analysis by incorporating the structure reconfiguration. In the scope of this research is the analysis of what SC elements will survive (i.e., be in operation) after a disruption and under which conditions (i.e., failure in a group of suppliers that interrupt the SC operation fully) the SC can still survive or it will lose its resilience.

The research methodology is based on a hybrid fuzzy-probabilistic approach to network resilience analysis. The SCD resilience analysis is performed as the analysis of optimistic, pessimis- The results have some major implications. First, it suggests a method to compare different SCDs regarding the resilience. Second, since the reconfiguration is included, the resilience analysis becomes more realistic and considers both disruptions and recovery. Third, the developed method can be used both for monotonous and non-monotonous structures. Fourth, the usage of minimum structure failure edge-cuts allows identifying groups of critical suppliers or a critical supplier whose failure separates the network into two disconnected parts and interrupts the SC operation fully.



  Resilience consideration without recovery measures and  Resilience consideration with recovery measures.

Fig. 1 .

 1 Fig.1. Disruption consideration without and with recovery measures

  then the structure is monotonous and for the failure poly- nomial is true (0) 1 T  and (1) 0 T  ;  , then the structure is non-monotonous and for the failure polynomial is true (1) 0 T  ; if 0 1

  of the network structure failure. According to reliability theory, in the function class ( ) L X a pos- sibility measure integral is defined in (Pyt'ev 2002). The function class ( ) L X includes the func- tions : f X L  , where ([0, 1], , , ) L     is the possibility value scale in the interval [0, 1], sub- ject to operators  , «+» and «  »:

  belongs to the class ([0, 1]) L .

  changes in the intermediate states  path ((i.e., the case where recovery actions are included) can be described as the following optimization problem (11

  puted. The random transition to an intermediate state subject to genome( )

  structure reconfiguration is shown.

Fig

  Fig. 4. Supply chain structure reconfiguration path

Fig. 5 .Fig. 6 .

 56 Fig. 5. References SC network structuresIn terms of the probabilistic-fuzzy approach, these structures are represented as shown in Fig.6.

  tic, and random (possible) reconfiguration paths of an SC structure due to disruptions. The application of the developed structure resilience index is demonstrated by the computation on the referenced SCD examples from the literature.

Table 1

 1 Computed values of SCD resilience

	Structure		SCD structural resilience SU J 
		Worst-case (no	Average case	Average case	( ) SU E J 
		reconfiguration)	(no	(with
			reconfiguration)	reconfiguration)
	a	0.0615	0.1548	0.7056	0.3073
	b	0.0496	0.2538	0.8606	0.3880
	c	0.0289	0.2527	0.8306	0.3707
	d	0.0456	0.1640	0.7519	0.3205
	In

Table 2 ,

 2 the results are compared with the resilience estimation from[START_REF] Kim | Supply network disruption and resilience: A network structural perspective[END_REF].

Table 2

 2 Comparison of the values of SCD resilience

	Structure		Resilience values
		Kim et al. (2015)	Our approach	Our approach (no
			(with reconfigu-	reconfiguration)
			ration)
	a	0.11	0.31	0.15
	b	0.30	0.39	0.25
	c	0.16	0.37	0.25
	d	0.13	0.32	0.16

In future, the gained insights require more detailed analysis efforts regarding the effect of the SCD structure reconfiguration as resilience increase driver. It is a very important task to develop methods of how to restore the SCD structure from an available genome. This task is quite similar to inverse problems in linear programming. In addition, the complexity issue in genome calculation (e.g., for structure C, the polynomial contains over 3,000 elements) needs to be addressed based on the development of efficient heuristic methods. Finally, development of risk mitigation policies for critical suppliers at the minimum edge-cuts is an interesting future research avenue.


Thomas Rulleau

Lucette Toussaint
email: lucette.toussaint@univ-poitiers.fr

Massage can (re-)activate the lower-limb sensorimotor representation of older adult inpatients

Keywords: Massage, Sensorimotor representation, Older adult, Lower limb, Mental rotation

Understand how changes in afferent signal processing may impact the sensorimotor processes is essential for physical therapists whose objective is to actively improve the reorganization of motor function in patients suffering from sensorimotor system disturbance.

Because the sensorimotor processes are slowed with the advance in age, we examined whether a single massage session can reactivate the sensorimotor processes of older adult inpatients.

Participants were randomly assigned to the experimental (with massage) or control (without massage) groups. Massage was realized on both feet with 7.30 minutes spent on each foot (Experiment 1), the right foot or the right foot and knee for 10 minutes (Experiment 2). Body and non-body mental rotation tasks were used to assess the lower-limb motor representation before (pre-test), immediately after (post-test 1) and 24 hours after the massage (post-test 2). Results

showed the positive impact of massage on the body mental rotation task. The activation of the sensorimotor processes can last up to 24 hours depending on the extent of the massaged area.

Importantly, the activation of the sensorimotor representation concerned not only the massaged leg but also the contralateral leg. No difference between groups appeared in the non-body mental rotation task which did not solicit the sensorimotor processes. These results highlighted that peripheral activation via a massage had a specific impact on the sensorimotor processes. Massage is an interesting technique which can help older adult inpatients cope with the slowdown of the signal processing related to advancing age.

Plasticity is an old concept in neuroscience that led many researchers to take an interest in the structural as well as the functional reorganization of the central nervous system according to environmental demands and experiences. Pascual-Leone and collaborators (2005) wrote in their review on the plasticity of the cortex of the human brain that "The challenge we face is to learn enough about the mechanisms of plasticity and the mapping relations between brain activity and behavior to be able to guide it, suppressing changes that may lead to undesirable behaviors while accelerating or enhancing those that result in a behavioral benefit for the subject or patient."

One topic of interest related to plasticity is understanding how changes in afferent or efferent signal processing may be observed at the behavioral level to enhance changes that can favor behavioral benefits and suppress those that can induce undesirable behaviors for the subject or the patient.

Over the past two decades, researchers in cognitive neuroscience have revealed that the humain brain is rich in reorganization and changes not only in younger but also in older adults.

These changes observed in various parts of the brain may affect cognitive and sensorimotor functions, aging being associated with the progressive loss of these functions. For example, sensorimotor changes with aging are highlighted by slower and less smooth movements [START_REF] Diggles-Buckles | Age-related slowing[END_REF][START_REF] Goble | Proprioceptive sensibility in the elderly: degeneration, functional consequences and plastic-adaptive processes[END_REF], increased spatial and temporal variability [START_REF] Contreras-Vidal | Elderly subjects are impaired in spatial coordination in fine motor control[END_REF], difficulties to perform multi-joint movements [START_REF] Seidler | Changes in multi-joint performance with age[END_REF], postural instability that increases the risk for fall [START_REF] Tinetti | Risk factors for falls among elderly in persons living in the community[END_REF]) and so on. The sensorimotor performance impairments with aging are likely due to changes in peripheral structures and central nervous system (see [START_REF] Seidler | Motor control and aging: links to age-related brain structural, functional, and biochemical effects[END_REF], for a review on aged-related differences and motor deficits in older adults). In the present study, we will focus on the positive effects of brain plasticity in older adults, especially considering the sensorimotor function and the effects of massage on the sensorimotor representation. This knowledge can be important for physical therapists whose objective is to actively improve the reorganization of motor function in patients suffering from sensorimotor system disturbances.

In cognitive neuroscience, some studies that focused on the plasticity of the sensorimotor system mainly used two approaches that were based on either a stimulating or impoverished environment to assess the continuous and rapid changes in sensorimotor representation. Some researchers using brain mapping techniques have revealed a decrease in motor cortex excitability [START_REF] Avanzino | Shaping motor cortex plasticity through proprioception[END_REF][START_REF] Facchini | Time-related changes of excitability of the human motor system contingent upon immobilization of the ring and little fingers[END_REF] as well as a disruption in motor performance [START_REF] Bassolino | Functional effect of shortterm immobilization: kinematic changes and recovery on reaching-to-grasp[END_REF]Huber, Ghilardi, Massimini, Ferrarelli, Reidner, Peterson, Tononi, 2006) following 10-12 hours to 4 days of immobilization of the fingers or an arm. Recently, to examine the central and functional effects of non-use of a limb, some researchers based their reasoning on the simulation theory [START_REF] Jeannerod | Neural simulation of action: a unifying mechanism for motor cognition[END_REF], which states that physical and simulated actions share the same sensorimotor representations and rely on similar mechanisms. The authors specifically examined whether internal sensorimotor representations are affected by the input/output restriction of signal processing following a short delay of upper-limb non-use (24 or 48 hours; [START_REF] Meugnot | Motor imagery practice may compensate for the slowdown of sensorimotor processes induced by short-term upper-limb immobilization[END_REF][START_REF] Meugnot | The embodied nature of motor imagery processes highlighted by short-term limb immobilization[END_REF] by asking participants to solve mental rotation tasks using body or non-body stimuli that depended respectively on motor and visual imagery strategies, respectively. Based on a motor imagery strategy, the mental rotation tasks used body images (i.e., hand or foot) to assess the efficiency of specific internal sensorimotor representation (i.e., the upper-limb or the lower-limb representation, respectively). The results showed that immobilized participants took more time than controls (i.e., non-immobilized participants) to solve the body (hand) mental rotation task (i.e., to identify whether the stimulus corresponds to either a left or a right hand), whereas no differences were detected between the two groups when solving the non-body mental rotation task (i.e., to identify whether the stimulus corresponds to the number "2" or its mirror image).

Moreover, a short period of sensorimotor restriction did not lead to a general slowdown in the sensorimotor processes. A somatotopic effect induced by 24 hours of left-hand immobilization have been reported, as revealed by longer response times for the stimuli depicting the immobilized hand compared to the non-immobilized hand [START_REF] Meugnot | The embodied nature of motor imagery processes highlighted by short-term limb immobilization[END_REF]2015). In contrast, 48 hours of left-hand non-use affected both the immobilized and the non-immobilized hand [START_REF] Toussaint | Short-term limb immobilization affects cognitive motor processes[END_REF]. Moreover, if 48 hours of left-hand immobilization impairs the effector-system corresponding to the restricted limb (i.e., the upper-limb system), it does not extend to another effector-system (i.e. the lower-limb system) [START_REF] Meugnot | Selective impairment of sensorimotor representations following short-term upper-limb immobilisation[END_REF].

Other researchers that have focused more on the effect of a stimulating environment on sensorimotor representations confirmed its central consequences. Neuroimaging studies have shown that enhancing sensory input by proprio-tactile stimulations (vibrations on muscles) modulates the excitability of the motor cortical projections to a specific limb and to the opposite limb [START_REF] Kossev | Crossed effects of muscle vibration on motor-evoked potentials[END_REF][START_REF] Rosenkranz | The effect of sensory input and attention on the sensorimotor organization of the hand area of the human motor cortex[END_REF] and can reduce the decrease in motor cortex excitability due to limb non-use [START_REF] Avanzino | Shaping motor cortex plasticity through proprioception[END_REF][START_REF] Roll | Illusory movements prevent cortical disruption caused by immobilization[END_REF]. From a behavioral point of view, the functional relevance of a stimulating environment has been demonstrated by experiments showing the effect of augmented sensory feedback on movement control in patients. For example, adding or enhancing proprioceptive information by muscle vibrations improved head and trunk movements in patients with torticollis [START_REF] Karnath | Effect of prolonged neck muscle vibration on lateral head tilt in severe spasmodic torticollis[END_REF], finger movements in pianists with musician's dystonia [START_REF] Rosenkranz | Regaining motor control in musician's dystonia by restoring sensorimotor organisation[END_REF] and gait control in Parkinson's disease patients following vibrations on lower limb muscles [START_REF] El-Tamawy | Effects of augmented proprioceptive cues on the parameters of gait of individuals with Parkinson's disease[END_REF]. Somatosensory stimulation can also be performed by physical therapists with a massage procedure. In particular, massage therapy is known to reduce pain and increase the range of motion in patients with knee arthritis pain (30 minutes/week for 4 weeks), improve active knee flexion after knee arthroplasty following one week of treatment (20 minutes/day) [START_REF] Field | Massage therapy research[END_REF], and improve physical fitness (strength, flexibility, agility, speed) in healthy soccer players (30 minutes every 3 days for 10 days) [START_REF] Hongsuwan | Effects of thai massage on physical fitness in soccer players[END_REF]. Sensory stimulation for 3 hours has been shown to enhance tactile acuity, haptic object exploration and fine motor control in the older adults [START_REF] Kalisch | Improvement of sensorimotor functions in old age by passive sensory stimulation[END_REF]. Moreover, some authors have shown that a 20minute "over-activation" of somatosensory information (by manual massage and mobilization of both the feet and ankles joints) allows the older adults to compensate for the absence of vision for regulate their postural sway [START_REF] Vaillant | Effect of manipulation of the feet and ankles on postural control in elderly adults[END_REF] as well as to improve functional balance performances (assessed by means of the One Leg Balance test and the Time Up and Go test) [START_REF] Vaillant | Effect of manipulation of the feet and ankles on postural control in elderly adults[END_REF][START_REF] Vaillant | Massage and mobilization of the feet and ankles in elderly adults: Effect on clinical balance performance[END_REF]. Even if these later experiments clearly showed that such a therapeutic intervention plays a major role in postural control, nothing is known about the importance of massage taken in isolation.

The functional changes that probably followed massage, although still poorly documented, could reveal the impact of a massage procedure on the sensorimotor system and legitimized intervention with massage by therapists in rehabilitation programs. However, the origin of these functional changes is unknown, and an important issue for cognitive scientists is to clarify whether these functional changes resulted from peripheral (i.e., the softening of muscles and/or joints) or central effects (i.e., the activation of the sensorimotor processes/representation).

Therefore, the aim of the present study was to examine the effect of a single massage session on mental rotation tasks using either body or non-body stimuli (Experiments 1 and 2), which was also performed in the study of the immobilization-induced effects on the sensorimotor representation [START_REF] Meugnot | The embodied nature of motor imagery processes highlighted by short-term limb immobilization[END_REF]2015;[START_REF] Toussaint | Short-term limb immobilization affects cognitive motor processes[END_REF]. Because aging disturbs the sensorimotor representation, as revealed by the slowdown in response times on mental rotation tasks with body parts [START_REF] Saimpont | Motor imagery and aging[END_REF][START_REF] Saimpont | Aging affects the mental rotation of left and right hands[END_REF], we chose to examine the effect of a single massage session on the mental rotation abilities of older adult participants. In Experiment 1, a physical therapist performed a massage on both feet successively for 15 minutes. In Experiment 2, the therapist performed massage for 10 minutes on either the right foot or the right foot and the right knee. We expected that the massage-induced effects on sensorimotor processes would be manifested by better performance (i.e., a decrease in response times) when solving the body mental rotation task (with foot images), such a result confirming the central effects of massage. The importance of the extent of the massage area on the activation of the sensorimotor processes was specifically investigated in Experiment 2 by comparing the effects of massage on the foot only versus the foot and the knee. In Experiment 2, we also investigated the bilateral activation of the sensorimotor system following unilateral massage (i.e., massage on one side of the body). In both experiments, no positive impact of the massage was expected in the non-body mental rotation task (with number images), which did not spontaneously elicit the use of sensorimotor processes [START_REF] Dalecki | Mental rotation of letters, body parts and complex scenes: Separate or common mechanisms ?[END_REF].

Experiment 1

Method Participants 32 right-handed inpatients voluntarily participated in the experiment (mean age 78.5 years, SD = 7.6 years). They were hospitalized for diverse geriatric or neurogeriatric reasons (asthenia, general state alteration, falls, chronic obstructive pulmonary disease, depression, etc.)

and were still at the hospital during the experiment. All participants were able to walk 10 meters in less than 30 seconds. They had normal or corrected-to-normal vision and provided written informed consent for their participation prior to their inclusion in this study. Before testing, the participants were randomly divided into 2 groups: a control group (n=16, mean age 79±7.5 years, 9 males) and a massage group (n=16, mean age 78 ±8.5 years, 8 males). The study protocol was in accordance with the ethical standards of the local ethics committee of the hospital center where the experiment occurred. All participants were naïve to the purpose of the experiment. However, they received both written and verbal descriptions of the experimental procedure and signed consent forms indicating agreement to participate in the experiment.

Tasks and material

All participants performed 2 mental rotation tasks using either body or non-body stimuli.

For both tasks, participants were seated in front of a computer screen (̴ 60 cm) and instructed to place their left and right index fingers on 2 marked keys located on the left and the right sides of the keyboard, respectively. Participants were asked to identify the images displayed on the center of the computer screen as quickly and accurately as possible. In the body mental rotation task, the stimuli consisted of pictures of right or left feet (created with Poser 6.0 software; sized 20.7 x 12.7 cm; Figure 1A). The participants had to determine the laterality of the foot images and answer by pressing the left-marked key for a left foot image or the right-marked key for a right foot image. In the non-body mental rotation task, the stimuli consisted of the number "2" or its mirror image (20.7 x 12.7 cm; Figure 1B). Participants had to determine whether the number was presented in its canonical form or its mirror image by pressing the appropriate left or right key.

For both tasks, the foot and number stimuli were presented in different orientations in the plane of the images (i.e., 0°, 40°, 80° and 120° in clockwise and counterclockwise directions). A trial began when a fixation cross was displayed in the center of the screen for 500 ms. Then, a stimulus was presented and remained visible until the participant provided his/her response. The E-Prime 2.0 software package (Psychology Software Tools Inc., Pittsburgh, USA) was used to present images and record the participants' responses (accuracy and response times).

Procedure

The participants were divided into 2 groups: a massage group and a control group. In the massage group, the intervention method was standardized. The massage technique consisted of effleurage and kneading and friction with moderate pressure applied under the foot (see [START_REF] Field | Massage therapy research[END_REF], for a review). The massage was performed by a physical therapist with 9 years of experience. Both feet were successively massaged for 7.30 minutes each. The control group did not undergo the massage procedure, but they talked (about their family, weather, etc.) with the therapist for 15 minutes.

The body and non-body mental rotation tasks were performed during three experimental sessions: before (pretest), immediately after (posttest 1) and 24 hours after the massage (posttest 2). For each session, each task was divided into 2 phases: the familiarization phase and the experimental phase. During the first familiarization phase, participants were shown 14 randomly presented trials (illustrated in Figure 1). During the second experimental phase, participants were shown 5 blocks of 14 trials (i.e., 70 trials per participants) presented in a random order within each block. In the three experimental sessions, the body mental rotation task was performed before the non-body mental rotation task because the sensorimotor processes may be attenuated when a non-body mental rotation task is performed first [START_REF] Toussaint | Short-term limb immobilization affects cognitive motor processes[END_REF].

Data analysis

Accuracy and response times were recorded and analyzed. Only data from correct responses were used to analyze response times. Separate ANOVAs were performed for the body mental rotation task and the non-body mental rotation task on accuracy (%) and response times (ms) with group (control vs. massage) as a between-subjects factor and session (pretest, posttests 1 and 2) and rotation (0°, 40°, 80° and 120°) as within-subjects factors. Preliminary analyses revealed similar results for clockwise and counterclockwise directions for both body and nonbody stimuli oriented to 40°, 80° and 120° angles, which lead us to average the data with the same rotation angles to increase reliability (for a similar procedure, see [START_REF] Wilson | The link between motor impairment level and motor imagery ability in children with developmental coordination disorder[END_REF]. Post hoc comparisons were carried out with Newman-Keuls test. Alpha was set at .05 for all analyses.

Results

The body mental rotation task ANOVA performed on the percentage of correct responses showed a mean effect of rotation only (F 3,90 = 27.09, p < .0001, ŋ² p = .47). Post hoc comparison revealed that correct responses were more frequent for the 0°, 40° and 80° foot rotations (M = 95 %; SD = 8%) than the 120° rotation (M = 82%; SD = 15%; ps<.0002), regardless of the group and the session.

ANOVA on the response times showed a mean effect of session (F 2,60 = 5.25, p < .008, ŋ² p = .15) and rotation (F 3,90 = 51.66, p < .0001, ŋ² p = .63), as well as a significant session x group interaction (F 2,60 = 3.59, p = .034, ŋ² p = .11). Post hoc comparisons showed that response times increased from the 40° to 120° foot rotations (40°: M = 1352 ms, SD = 149 ms; 80°: M = 1489 ms, SD = 162 ms; 120°: M = 1740 ms, SD = 190 ms; ps < .006), while no significant differences were for the 0° to 40° rotations (0°: M = 1298 ms, SD = 131 ms; p < 0.27). As illustrated in Figure 2 and confirmed by post hoc comparisons, response times decreased from the pretest to the posttests following the massage (ps < .015) without a distinction between posttests 1 and 2 (p = 0.62). No significant differences were observed between the pretest and posttests for the control group (ps > 0.41). Moreover, response times were shorter in posttests 1 and 2 for the massage group than for the control group (ps < .05), while no significant difference appeared in pretest.

Discussion

The aim of the first experiment was to investigate the influence of massage on the mental rotation of body stimuli as an indicator of the efficiency of the sensorimotor processes. We expected that enhancing sensory input by means of a massage procedure performed by a physical therapist would activate the sensorimotor processes in older adult participants. For this purpose, we evaluated participants at pretest (i.e., before massage), posttest 1 (i.e., immediately after massage) and posttest 2 (i.e., 24 hours later). A similar pretest/posttest procedure was used for the control participants who did not undergo the massage procedure. The main results of Experiment 1 showed that the response times in the body mental rotation task significantly decreased after 15 minutes of a foot massage, while no significant differences were detected in the non-body mental rotation task between the pretest and posttests. The changes in response times reported for the body mental rotation task cannot be explained by a trade-off with response accuracy, as the percentage of correct responses did not differ between the groups regardless of the session and the rotation angles of the foot images. Importantly, the improvement in response times following the massage was maintained in the posttest performed after a 24-hour delay (in posttest 2). These findings revealed that visual imagery performance, which was evaluated with the non-body mental rotation task, did not show any effect related to the limb-massage procedure. In contrast, motor imagery performance, which was evaluated with the body mental rotation task, was improved by a single and brief massage session (15 minutes) performed by a physical therapist with 9 years of experience. The performance improvement was not manifested by an increase in the success of the task (i.e., the percentage of correct responses) but by the activation of the sensorimotor processes required to solve the task. Therefore, unlike the immobilization procedure which showed the negative effect of an impoverished environment on the sensorimotor representation with the slowing of the sensorimotor processes induced by input/output restriction [START_REF] Meugnot | The embodied nature of motor imagery processes highlighted by short-term limb immobilization[END_REF][START_REF] Toussaint | Short-term limb immobilization affects cognitive motor processes[END_REF], the present experiment highlighted the positive effect of a stimulating environment. Enhancing sensory input by massage led to rapid updates to the sensorimotor representation that may be more effective or easier to access due to an increase in proprioceptive signals. Similar observations were previously reported specifically with a vibratory stimulation procedure that activates the sensorimotor-related area [START_REF] Naito | Kinesthetic illusion of wrist movement activates motor related areas[END_REF][START_REF] Romaiguere | Motor and parietal cortical areas both underlie kinaesthesia[END_REF]. The present experiment does not cast doubt on the peripheral effects of massage (on muscle stiffness and peripheral blood flow; [START_REF] Liu | Recent advances in massage therapy -a review[END_REF] but shows, for the first time in the literature, the cognitive (or central) effects of a massage procedure that activates the sensorimotor processes. Further experiments will be carried out to determine whether the positive effect of massage on the processing of sensorimotor information is accompanied by a significant improvement in functional balance performances.

Importantly, the comparison between posttests 1 and 2 in the body mental rotation task revealed that the positive effect of massage on the sensorimotor processes was also found 24 hours after the intervention. These findings are interesting because they showed that a single and brief massage session performed by a physical therapist improved the functioning of the sensorimotor system and revealed that the massage is still effective one day later. However, the present experiment did not provide information on the importance of the extent of the massage area on the activation of sensorimotor processes or the duration of the massage-effect as a function of the extent of the massage area. These points were specifically investigated in the following experiment.

Experiment 2

The second experiment aimed to replicate the positive effects of massage on sensorimotor processes, which were shown in Experiment 1, and investigate the importance of the extent of the massage area on the activation of sensorimotor processes, as well as the possibility that these peripheral activations can activate not only the sensorimotor representation of the massaged limb but also of the opposite limb. It is particularly important at the theoretical level but also at the practical level to know whether contralateral activation of the sensorimotor system could be observed following the massage of a specific limb. For example, a positive massage-induced effect on the contralateral limb could aid in the rehabilitation of an immobilized limb that is still in a cast because of a fracture by regularly reactivating the sensorimotor processes of the nonused limb.

Method

Participants 34 right-handed inpatients voluntarily participated in the experiment (mean age = 77 years, SD = 8.8 years). None of them participated in experiment 1. They were hospitalized for diverse geriatric or neurogeriatric reasons (asthenia, general state alteration, falls, chronic obstructive pulmonary disease, depression, etc.) and were still at the hospital during the experiment. All inpatients were able to walk 10 meters in less than 30 seconds. They had normal or corrected to normal vision and provided written informed consent prior to their participation and inclusion in the study. Before testing, participants were randomly divided into 2 groups: a foot massage group (n = 17, mean age = 75 years, SD = 9.2 years, 6 males) and a foot-knee massage group (n = 17, mean age = 79 years, SD = 8.1 years, 7 males). The study protocol was in accordance with the ethical standards of the local ethics committee of the hospital center where the experiment occurred. All participants were naïve to the purpose of the experiment. However, they received written and verbal descriptions of the experimental procedure and signed consent forms indicating agreement to participate in the experiment.

Tasks and material

Participants performed 2 mental rotation tasks using either body or non-body stimuli.

Both tasks were similar to those used in Experiment 1.

Procedure

The participants were divided into 2 groups: a foot massage group and a foot-knee massage group. In the foot massage group, the massage was performed for 10 minutes on the right foot of each participant. In the foot-knee massage group, massage was successively performed for 5 minutes on the right foot and 5 minutes on the right knee of each participant. In both groups, the massage technique was similar to that used in Experiment 1 and consisted of effleurage and kneading and friction with application of moderate pressure. Massage was performed by the same physical therapist who took part in the first experiment.

Similar to Experiment 1, the body and non-body mental rotation tasks were performed during three experimental sessions: before (pretest), immediately after (posttest 1) and 24 hours after the massage (posttest 2). For each session, participants practiced 2 phases (the familiarization and the experimental phases), and the body mental rotation task was performed before the non-body mental rotation task.

Data analysis

Accuracy and response times were recorded and analyzed. Only data from correct responses were used to analyze response times. Separate ANOVAs were performed for the body and the non-body mental rotation tasks. For the body mental rotation task, ANOVAs were performed on accuracy (%) and response times (ms) with group (foot massage vs. foot-knee massage) as a between-subjects factor and session (pretest, posttest 1, posttest 2), foot (right vs. left) and rotation (0°, 40°, 80° and 120°) as within-subjects factors. For the non-body mental rotation task, ANOVAs were performed on accuracy (%) and response times (ms) with group (foot massage vs. foot-knee massage) as a between-subjects factor and session (pretest, posttest 1, posttest 2) and rotation (0°, 40°, 80° and 120°) as within-subjects factors. Post hoc comparisons were carried out with Newman-Keuls test. Alpha was set at .05 for all analyses.

Results

The body mental rotation task

ANOVA on the percentage of correct responses showed only a mean effect of rotation (F 3,96 = 35.78, p < .0001, ŋ² p = .53). Post hoc comparisons revealed that correct responses were significantly more frequent for the 0°, 40° and 80° rotations (M = 95%; SD = 8%) than the 120° rotation (M = 89%; SD = 13%, ps <.0001), regardless of the group, the foot and the session.

ANOVA on the response times showed mean effects of session (F 2,64 = 17.98, p < .0001, ŋ² p = .36) and rotation (F 3,96 = 39.18, p < .0001, ŋ² p = .55), as well as a group x session interaction (F 2,64 = 3.24, p < .04, ŋ² p = .10). As illustrated in Figure 3 and confirmed by post hoc comparisons, response times significantly decreased from pretest to posttest 1 for both groups (ps < .01) and from pretest to posttest 2 (i.e., after a 24-hour delay) in the foot-knee massage group only (p < .001). No effect of the foot (right or massaged-foot vs. left or non-massaged foot) was reported. To quantify how the massage improved the response times immediately after (in posttest 1) and 24 hours later (in posttest 2), we computed the Index of Performance Improvement (IPI=[response time in posttest-response time in pretest]/response time in pretest, expressed as a percentage) for each participant. A positive value indicated a performance improvement (i.e., a decrease in response time in the posttest), whereas a negative value indicated a performance deterioration (i.e., an increase in response time). IPI was analyzed by ANOVA with group (foot massage vs. foot-knee massage) as a between-subjects factor and posttest (posttest 1 vs. posttest 2) as a within-subjects factor. T-tests were used to examine whether the IPI significantly differed from zero.

The ANOVA revealed only a significant group x posttest interaction (F 1,32 = 3.91, p < .05, ŋ² p = .09]. Post hoc comparisons revealed that performance improvement from pretest to posttest 1 (i.e., immediately after the massage) was similar in both groups, whereas the performance improvement from pretest to posttest 2 (i.e., 24 hours after the massage) was better in the footknee massage group than the foot only massage group (p < .02; Figure 4). Moreover, in the foot massage group, the IPI was significantly smaller in posttest 2 than posttest 1 (p < .05). T-test analyses revealed that the IPI was significantly different from zero in posttest 1 for the foot massage group (t 17 = 5.30, p < .0001) and the foot-knee massage group (t 17 = 5.11, p < .0001), as well as in posttest 2 for the foot-knee massage group (t 17 = 5.70, p < .0001), whereas the IPI did not differ from zero in the foot massage group (t 17 = 1.43, p = .17).

The non-body mental rotation task

ANOVA on the percentage of correct responses showed only a main effect of rotation (F 3,96 = 19.01, p < .0001, ŋ² p = .37). Post hoc comparisons revealed that correct responses were significantly more frequent for the 0° and 40° rotations (M = 93%; SD = 15%) than the 80° (M = 89%; SD = 18%, p < .001) and 120° rotations (M = 82%; SD = 23%, p < .0001), as well as the 80° compared to 120° rotation (p < .001).

ANOVA on the response times showed only a mean effect of rotation (F 3,96 = 11.04, p < .0001, ŋ² p = .27). Post hoc comparisons revealed that response times were significantly lower for the 0° and 40° rotations (M = 1128ms; SD = 150 ms) than the 80° (M = 1263 ms; SD = 162 ms, < .001) and 120° rotations (M = 1492ms; SD = 206 ms, p < .0001), as well as for the 80° and 120° rotations (p < .0001), regardless of the group and the session.

Discussion

The second experiment had 2 specific aims. First, it was carried out to examine whether the extent of the massage area (i.e., massage performed on the foot only or on the foot and the knee of the right leg) differentially impacted the reactivation of the sensorimotor processes.

Second, it was carried out to examine whether contralateral activation might be observed following massage performed for 10 minutes on the right leg (i.e., activation of the sensorimotor representation of the left leg not massaged by the physical therapist). With the exception of the massage, the pre/posttest experimental procedure was similar to Experiment 1. The main results of Experiment 2 revealed that, for the body mental rotation task, response times similarly decreased from pretest to posttest 1 in both groups (the foot massage group and the foot-knee massage group), but differences between groups appeared in posttest 2, with better improvement in response times following a 10-minute massage on both the foot and the knee. Importantly, the positive effects of massage were detected for both the right foot (the massaged foot) and the left foot (the non-massaged foot) regardless of the posttest (i.e., immediately after the massage and 24 hours later). No difference between groups appeared in the non-body mental rotation task.

The results of Experiment 2 confirmed those of Experiment 1, in particular the positive effect of a single and brief (10 minutes) massage session performed by an experienced physical therapist on the activation of the sensorimotor processes. Moreover, when the massage-induced effects were evaluated immediately after the massage session, no difference appeared as a function of the extent of the massage area. The participants actually took less time to identify the laterality of foot images when the task was performed after the massage in both groups (foot massage and foot-knee massage). However, the present experiment showed that the extent of the massage area was important when the effects of massage were assessed 24 hours later. In that case, the activation of the sensorimotor processes lasted longer when the massage area was more extensive. Therefore, activating a larger body representation via a massage procedure on both the foot and the knee during a brief session is better than concentrating the massage on a specific area (the foot only) during the same period. This suggests the need to diversify the location of the massage to stimulate the sensorimotor system in a sustainable manner.

In Experiment 2, massage performed on the right leg alone activated not only the sensorimotor representation of the right leg but also that of the opposite leg. The results showed that improvement in response times following the massage was similar regardless the laterality of the foot that needed to be identified. These findings revealed that, for a given effector system (the lower-limb in the present experiment), massage on one side of the body activated the sensorimotor representation of the opposite side. Similar results were previously observed following the restriction of input/output of signal processing via an immobilization procedure [START_REF] Meugnot | Selective impairment of sensorimotor representations following short-term upper-limb immobilisation[END_REF][START_REF] Toussaint | Short-term limb immobilization affects cognitive motor processes[END_REF], which resulted in a slowdown of the sensorimotor processes for both the immobilized and the non-immobilized effectors. In the same vein, other studies have reported a positive transfer of a learned task to the contralateral side of the body [START_REF] Vangheluwe | Learning and transfer of an ipsilateral coordination task: Evidence for a dual-layer movement representation[END_REF]. In all cases, it may be that the interhemispheric exchange of sensorimotor information was made possible by the corpus callosum, which is considered a central structure in hemisphere interconnection processes [START_REF] Franz | Dissociation of spatial and temporal coupling in the bimanual movements of callosotomy patients[END_REF]. In the present experiment, we clearly showed that the central consequence of the enhancement of sensory input by unilateral massage was the induction of both ipsilateral and contralateral activation of the sensorimotor system, which supports the existence of an effector-independent sensorimotor representation [START_REF] Vangheluwe | Learning and transfer of an ipsilateral coordination task: Evidence for a dual-layer movement representation[END_REF]. For physical therapists, the bilateral activation reported in the present experiment is very interesting because it demonstrates the possibility of using massage on one side of the body alone in case of contraindications, e.g., when an older adult is wearing a cast due to a fracture (of the hand, of the foot) or when he experiences severe pain with touch, limb manipulation on one specific side of his body.

Conclusion

The present 2 experiments offer support for the hypothesis that a single and brief massage session performed by an experienced therapist has functional effects. Massage activates the sensorimotor receptors that favor the functioning of the related sensorimotor system. The positive impact of massage is not only immediate but lasts up to 24 hours later. Importantly, the activation of the sensorimotor representation may partially depend on the extent of the area and may concern not only the massaged leg but also the contralateral leg.

Overall, these findings confirmed that peripheral activation via a massage had a specific impact on the sensorimotor processes and can help patients cope with the slowdown of the signal processing related to advancing age [START_REF] Saimpont | Aging affects the mental rotation of left and right hands[END_REF]2013). These findings highlighted the impact that massage therapies could have on geriatric care, in particular during programs for the prevention of falling or rehabilitation of autonomy. However, further studies should be carried out to examine whether the possibility of modulating the sensorimotor cortex activity by manipulating sensory inputs via massage is accompanied by improvements in motor function or produces specific improvements in rehabilitation (i.e., better motor performance for inpatients or quicker improvements). It may be that unlike immobilization, which induces a slowdown of sensorimotor processes and a deterioration in motor performance [START_REF] Bassolino | Functional effect of shortterm immobilization: kinematic changes and recovery on reaching-to-grasp[END_REF]Hubert et al., 2006), the massage-induced effect (i.e., activation of sensorimotor processes) is accompanied by an improvement in motor performance for older adults.

Finally, in the context of evidence-based practice, the present experiment showed that using body mental rotation tasks can be an adequate tool to objectively assess the central improvement of motor function following various rehabilitation programs in patients experiencing from sensorimotor system disturbances. Examining response times when identifying body-parts stimuli allows the physical therapist to objectively know the central effects of rehabilitation, i.e., whether specific sensorimotor representations of patients have improved or not, or whether other rehabilitation sessions or techniques are required to maximize the activation of the sensorimotor processes. Although further studies are needed, it is possible that the activation of the sensorimotor processes favors motor control in older adults inpatients.

Figure 1 .

 1 Figure 1. Illustration of the stimuli used in Experiments 1 and 2 for the body mental rotation task

Figure 2 .

 2 Figure 2. Mean response times (ms) for the body mental rotation task as a function of group

Figure 3 .

 3 Figure 3. Mean response times (ms) for the body mental rotation task as a function of group (foot

Figure 4 .

 4 Figure 4. Index of Performance Improvement (%) as a function of group (foot massage vs. foot-