The Study of Resilience of Transport and Logistics Systems Alexander N. Pavlov Dmitry A. Pavlov Valentin N. Vorotyagin Mozhaisky Military Aerospace Mozhaisky Military Aerospace Mozhaisky Military Aerospace Academy Academy Academy St. Petersburg, Russia St. Petersburg, Russia St. Petersburg, Russia pavlov62@list.ru dpavlov239@mail.ru vorotyagin@rambler.ru property of the system to preserve and restore its characteristics (vector quality indicator of the functioning of the TLS) under the influence of a catastrophic environment on the production and logistics process. To Abstract assess the resilience of a TLS taking into account the risks of failures in the event of design abnormal situations or Analysis of modern methods of evaluation of “normal” operating conditions, as a rule, a deterministic resilience of transport and logistics systems (TLS) approach is used, methods of reliability theory and in the management of their configuration and simulation modeling [Fox00, Rob02), Iva13, Mun15, reconfiguration under conditions of destructive Das15, Iva16, Kim15, Sim14, Xu14, Sny16]. The imitation effects has shown that in the design and creation of of TLS production and logistics processes is performed. TLS it is necessary to develop conceptually new The imitation of TLS elements, key nodes and connections methodological approach to the detection of failures is also produced. The failure of every disruption scenarios, recovery paths in TLS and aforementioned part leads to loss of the TLS resilience, carry out analysis of such important property of which depends on the modelled level of reliability. For each TLS as structural resilience of their configuration. time point of imitation, a functional check of the TLS The outcomes of this research constitute a useful functional elements is performed. The random time of decision-making support tool that allows detecting forced breaks in the work of one or another TLS node, the disruption scenarios at different risk-aversion values of the target indicators are estimated in case of levels based on the quantification of the structural failure. The calculation is terminated in case of failure of robustness with the use of the genome method and the TLS elements, in which further operation is impossible observing the scope of disruption propagation. Our (the occurrence of critical failures). Such calculations are results can be of value for decision-makers to performed for different levels of reliability of compare different TLS structural designs computational emergency situations. At each level, a regarding the robustness and to identify disruption predetermined number of statistical tests or an amount that scenarios that interrupt the TLS operations to provides the specified simulation accuracy is produced. The different extents. calculated data are displayed on the radar chart (Kiviat diagram). To determine the TLS resilience index, the area of the figure in the chart is compared with the areas of the 1 Introduction figures reflecting the assumed and admissible limit values of the target indicators. If at least one of the targets is less The structural TLS design may change due to disruptions, than the admissible limit value, then it corresponds to the defined as “events that interrupt the regular flow of goods loss of the TLS resilience, which requires a decision on the or services within a system” [Bla11]. Modern TLSs have nature of its further functioning. grown in scale and complexity, increasingly exposing firms But at present such dependencies are obtained only as a to various and scattered disruptive events [Hos16, Mis16, result of the exploitation of existing TLS. It’s a problem Iva18, Dub19]. The creation of effective TLS is possible by with the mentioned approach. But for new TLS design the ensuring their reliability and resilience both in nominal existing networks statistics is usually used. It is normal if conditions of operation and in the event of predictable and the new network is similar in structure and composition unpredictable disruptions. TLS resilience has become one with the previous TLS. But, if the developed TLS differs of the main research categories over the past decade significantly from the previously created ones, this [Gun15]. Moreover, the resilience is understood as the approach is not always acceptable. In addition to the predictable disruptions, there are Copyright c by the paper's authors. Use permitted under Creative unpredictable, such that no one can foresee in advance, and Commons License Attribution 4.0 International (CC BY 4.0). In: A. Khomonenko, B. Sokolov, K. Ivanova (eds.): Selected Papers of the therefore it is impossible to prepare for them in advance. Models and Methods of Information Systems Research And not least in real conditions of operation, these Workshop, St. Petersburg, Russia, 4-5 Dec. 2019, published at unpredictable disruptions occur, if not more often, then, at http://ceur-ws.org least, in frequency, they appear commensurate with the calculated ones. Under these conditions, models and 85 methods used in the theory of reliability, simulation certain sets of vertices; the number of vertices in the largest modelling are not applicable to ensure the TLS resilience, component of the graph is less than some predetermined which requires the development of a conceptually new number; the shortest path exceeds a given value. approach to ensuring the TLS resilience. Accordingly, the TLS is considered to be tenacious if these conditions are not met. 2 The Traditional Approach to the To analyze the properties of the structural resilience of the Assessment of the Structural TLS Resilience TLS under these conditions, as well as to synthesize a in the Conditions of Destructive Influences system with the required property of structural resilience, it is necessary to introduce a quantitative assessment that Within the framework of studies devoted to the adequately depicts the property in question. development of methodological foundations for ensuring When studying the TLS structural resilience according to the TLS resilience, it is necessary to analyze such an the approach proposed in that study [Pav18], introduces the important feature as the TLS configuration structural resilience. In a broad sense, the structural TLS resilience is notion of generalized failure of the i multiplicity, which understood to be such an ability of the object in question, considers the structural states of the TLS formed upon the which allows it to maintain, within certain limits, the quality sequential refusal of various combinations ( С ni ) of the of its target functioning (or restore such ability) by changing entire set of functional elements structures for i different (forming) the corresponding structures (configurations). functional elements ( i £ n where n is the number of The change in the structural states of the TLS is associated functional elements of the TLS structure considered). both with the proliferation and restoration of malfunctions Among the set of structural states for a given generalized in the elements of the structure of the TLS, and in the failure is determined by the set of working states, the power process of fulfilling orders. We will consider the failure (inoperable) the TLS functional element, which is not able of which we denote Ri , or the set of unworkable states, the to perform all the production and technological operations power of which we denote assigned to it. A functional element will be considered N i ( N i + Ri = Сni ). partially efficient if it can perform at least one of the assigned production and technological operations. It is For comparison of various structures, the relative function obvious that the values of the particular indicators of the of the TLS structural resilience is determined Y ( i ) quality of functioning of the TLS in each state depend on: n many failed, workable or partially workable functional Ri N elements; distribution of production and technological ( Y (i ) = Gi = i = 1 - ii ), its linear Сn Сn operations; reallocation of these operations between workable or partially workable functional elements. interpolation is performed by a piecewise linear function An important and indispensable condition for studying the ! ( x), x Î [0,1] Y and the integral indicator of the capabilities of the TLS is the analysis and evaluation of the structural resilience of the the TLS is introduced as the architecture of its structural states, reflecting both the 1 ! ( x)dx . functional and production-technological features of the TLS control. following functional Fg = Y ò 0 Structural models of the functioning of most complex technical systems can be correctly described [Rya76, We assume that the TLS is in an inoperable structural state Kop10, Pav18] by block diagrams, fault and event trees, if, in a generalized refusal, all elements that are included at connectivity graphs, multi-terminal networks, etc. least in at least one of the minimal failure sections of the However, these structural models can describe the TLS structure are removed. functioning of only monotonic systems. In monotonous In the most general case, the TLS structure is characterized models, it is impossible to take into account the logically by k minimal failure sections, each of which consists of complex and contradictory relationships and relationships m j ( j = 1,..., k ) elements. Moreover, the failure sections between functional elements, for example, which in some have common elements. structural states of the system increase, and in others, In this situation, the number of inoperable structural states decrease the indicator of the effectiveness of its functioning. with a generalized failure of the i multiplicity takes the Also, monotonous models do not represent systems in which elements simultaneously operate, some of which following form [Pav18a]: provide an increase, for example, reliability or resilience, and another part causes failures or accidents, i.e. has the opposite, detrimental effect on the security of the system as a whole. In the study of the TLS resilience, the structure of which is described by graphical models (monotone system [Pav18a], the TLS is considered “destroyed” if, in the case of deleting vertices or edges, the graph will satisfy one or several of the following conditions: the graph consists of at least two connected components; there are no directed paths for 86 k logical conditions for the implementation of its own N i = å d (i - m j )Cn - mjj - i-m functions by the elements and subsystems of the TLS. The j =1 second important aspect of building and further using the k k functional integrity scheme is an indication of the specific -å å d (i - m j1 - m j2 + m j1 j2 )Cn - mjj1 - mjj2 + mjj1 jj2 + i-m -m +m purpose of the simulation — the logical conditions for the 1 2 12 j1 =1 j2 > j1 realization of the system property being investigated, for k k k example, reliability or failure of the TLS, etc. + å å å d (i - m j1 - m j2 - m j3 + m j1 j2 j3 ) × , (1) It is known that the genome structure j1 =1 j2 > j1 j3 > j2 χ = ( c0 , c1 , c 2 ,..., c n ) [Pav18a], which is a i-m -m -m +m ×Cn - mjj1 - mjj2 - mjj3 + mjj1 jj2 jj3 - ... concentrated representation of the structural state of the 1 2 3 12 3 object, contains and allows to determine the following ...(-1) k -1d (i - m j1 - m j2 - ... - m jk + m j1 j2 ... jk ) × information in the process of structural study of complex i - m - m -...- m + m objects: first, information about the topological properties ×Cn - mjj1 - mjj2 -...- mjjk + mjj1 jj2 ...... jjk of the structure of a monotone system; secondly, 1 2 k 12 k information on the belonging of the object under study to ì1, x ³ 0 the class of monotone or non-monotonic systems; thirdly, where d ( x) = í is a discrete form of the to assess the indicators of the structural and functional î0, x < 0 resilience of the system. Heaviside step function. For the formal description and analysis of the process of In formula (1), the values m j j ... j represent the total degradation (restoration) of the TLS, we will consider the 1 2 k number of common elements in the minimum sections of operation of removing (restoring) critical elements failures with numbers j1 , j2 ,..., jk . { Pj , Pj , ..., Pj } = P! from the functional integrity scheme 1 2 N Using formulas (1), it is possible to calculate the relative as factors for changing the structure. In the general case, all function of the TLS structural resilience with a monotonic TLS functional elements can be considered as critical structure, and accordingly determine the integral index of elements. 1 In the process of removing (restoring) elements, the TLS ! ( x )dx . the structural resilience of the system Fg = Y ò 0 structure can be in one of its intermediate states Sa . According to the concept of the genome structure, structural To calculate the structural vitality, a set of minimum failure sections is needed, as well as the definition of common states Sa (initial, final, intermediate) are characterized by ! ! functional elements in these sections. In general, finding the their genomes ca ( ca by this material we mean the dual minimum failure rates is NP difficult. In this case, the analogue of the genome), while the indicators of the TLS calculation of the index of structural resilience using the structural and functional resilience, consisting of generalized formula (1) is a super-complex combinatorial homogeneous, non-uniform functional elements, depend on problem. At the same time, it should be noted that not all the reliability of their functions, can be calculated by the monotonic structures can be described using graphical following formulas [Pav18a]: models. ! ! 1 1 1 T Fhom ( ca ) = ca × (1, , ,..., ) , 2 3 n +1 3 The Genome Concept to the Assessment of ! ! 1 1 1 (2 the TLS Structural and Functional Resilience Fhet ( ca ) = ca × (1, , 2 ,..., n ) T , ) 2 2 2 in Conditions of Destructive Influences ! ! Fpossib ( ca ) = sup min{ca × (1, µ , µ ,..., µ ) , g ( µ )} 2 n T µÎ[0,1] To overcome the above features of estimating the TLS structural resilience, the following approach is proposed We assume that the structural state Sa characterized by the based on the concept of the genome structure [Kop10]. As ! a rule, the structural analysis of the functioning of a genome ca is directly related to the structural state S ! complex object begins with the construction of its described by the genome c , if there is a functional element functional integrity scheme (FIS) [Kop10], Pav18]. The ( $ Pj Î P! ), the failure (restoration) of which ( Pj = 0 or functional integrity scheme is a logically universal graphical tool for the structural representation of the studied Pj = 1 ) takes the system from state S to state Sa (from properties of system objects. The functional integrity schemes allow to correctly represent both all traditional state Sa to state S ). types of structural schemes (flowcharts, failure trees, event Let us designate this variation of the structural state of the trees, graphs of connectedness with cycles) and a ! P ! fundamentally new class of non-monotonic (non-coherent) PLS as follows: c ¬¾ j ® ca . The set of all structural ! structural models of various properties of the systems under states directly associated with the state c is denoted by study. The development of the TLS functional integrity ! schemes means, first of all, a graphical representation of the X (c ) . 87 One of the possible trajectories of the reconfiguration of the It should be noted that the maximum value of the TLS structure during the occurrence of failures (recovery) generalized index of structural and functional resilience can be described by the following chain of transitions J max = max{J k } will be achieved in the optimistic ! Pj1 ! Pj2 ! Pj3 k ca ¬¾® ca ¬¾® ca ¬¾® ... 0 1 2 scenario of reconfiguration of the TLS, and the minimum PjN -1 ! PjN ! value J min = min{J k } - in the pessimistic one. We will ... ¬¾¾ ® ca N -1 ¬¾¾ ® ca N , k ! ! ! ! Where ca 0 = c 0 , c a N = c f , the set conduct M simulation experiments. On each k experiment, a sequence is constructed { Pj , Pj , ..., Pj } = P! , i.e. the set of failed (restored) ! ! ! ! ! 1 2 N µV( k ) = éë ca , ca( k ) , ca( k ) , ..., ca( k ) , ca ùû (where element TLS in the transition chain is a permutation of the ! ! ! 0 ! 1 2 N -1 N elements of the set P! . ca = c 0 , ca = c f ) corresponding to 0 N the TLS The structural changes occurring in the intermediate state reconfiguration trajectory. For the constructed trajectory, ! ca on the reconfiguration trajectory will be evaluated by the value of the generalized index of structural and k one of the indicators of the structural and functional functional resilience J k = S 0 k is calculated. Next, we resilience of the TLS (2) included in the considered set: S ! ! ! ! find the average value of the structural resilience of all tests Ffailure ( ca )Î {Fhot ( ca ) , Fhet ( ca ), Fpossib ( ca )} . In ! 1 M k . Then it can be argued that the real values of addition, in each intermediate structural state ca , the TLS J0 = åJ M k =1 is characterized by a certain set of structural and topological the generalized index of the TLS structural and functional ! constraints Y l ( ca ) £ 0, l = 1, 2,..., L , formally defined resilience J SG are in the interval [ J min , J max ] and the most and quantified using (Pavlov et al. (2018)) relevant 0 expected value is J . In this case, the predicted values of indicators of structural vitality, flexibility, reachability, structural complexity, etc. In other words, these restrictions the indicator J SG can be set with a fuzzy triangular number define the range of allowable variations, which will be ( a, a , b ), where a = J 0, a = J 0 - J min , denoted in the following X . Then the task of building an optimistic (pessimistic) PLS b = J max - J 0 . reconfiguration scenario can be represented as the In addition, the calculation of the values of the structural following optimization problems (3). and functional resilience index N ! ! ! ! ! Ffailure ( ca )Î {Fhot ( ca ) , Fhet ( ca ), Fpossib ( ca )} can åF j =0 failure ( ca j ) ® ! max! (min) ca j ÎX ( ca j -1 ) (3) be made on the assumption that the TLS structure consists ! ! ! ! ca0 = c 0 , ca N = c f , only of elements that are homogeneous in the reliability of ! Y l ( ca j ) £ 0, l =1,2,..., L their functions, only elements that are not uniform in the { Pj1 , Pj2 ,..., PjN }= P" reliability of their functions, and finally there are potential In the work [Pav18a], a combined method of random failures to perform their functions. For each of these three directional search for solutions to the problem is cases, by calculating the indicator values J SG , we obtain, substantiated and an algorithm is developed that implements the above method. The combined method and respectively, three fuzzy triangular results: ( a о , a о , b о ),( the corresponding algorithm allows you to search for both optimistic and pessimistic trajectories, as well as a n , a n , b n ),( ab , a b , b b ). Then, as the value of the intermediate trajectories chosen randomly. generalized indicator of the TLS structural and functional Then, as a generalized indicator of the TLS structural and resilience J SG , we will assume the average value of the functional resilience, in the process of its structural results obtained reconfiguration according to the scenario µV( k ) , a (a , a , b ) + (a , a , b ) + (a , a , b ) . о о о n n n b b b J SG = relationship can be proposed S0k . Here 3 J = k Sk Thus, the task of calculating the value of the generalized ! (k ) ! (k ) failure ( ca j ) + F failure ( ca j +1 ) N -1 F indicator of the structural and functional resilience of the S0k = å , it is equal to the TLS TLS has been reduced to the analysis of optimistic, j =0 2 pessimistic or random (arbitrary) trajectories of the total structural and functional resilience functioning in the structural and functional reconfiguration of the object, process of reconfiguration within the scenario µV( k ) , and caused by failures (restoration) of the TLS functional ! elements. S k = max {Ffailure ( ca( kj ) )}" N is proportional to the It should be noted that the failure (recovery) of an element j = 0,1,..., N leads to the failure (recovery) of the remaining TLS TLS total structural and functional resilience functioning functional elements logically associated with it. Therefore, along the trajectory if the possible maximum resilience of in addition to the introduced generalized indicator of the the function is maintained during the development of the considered scenario. TLS structural and functional resilience J SG , it is possible 88 to introduce an absolute index of the TLS structural and represents the TLS (i.e., the graph G= (V, E)) as a network functional resilience. Each trajectory of the reconfiguration of non-disrupted and disrupted elements. Since the of the TLS structure is characterized by the number of structural genome represents the TLS design, each degradation levels J D , the last of which corresponds to the structural state Sa can be described by a genome ca . transfer of the TLS to an inoperable state. So for a Therefore, the total robustness or total failure of a path in pessimistic trajectory the number of levels is minimal and the TLS structure dynamics can be computed using Eqs. (2). equal J Dmin , for an optimistic trajectory it is maximal - J Dmax . The values of the absolute indicator of the TLS structural and functional resilience J AG will lie in the min max interval [ J D D , J ], and you can also calculate the most expected value equal D0 . In this case, the values of the J indicator J AG are similar, as well as J SG , can be set with a) a fuzzy triangular number ( aA , a A , b A ), where a A = J D0 , a A = J D0 - J Dmin , b A = J Dmax - J D0 . 4 Numerical example We explain the major determinants of the proposed method using an example. Consider an TLS given in Figure 1. b) c) Figure 2: Structural robustness and disruption scenarios a) pessimistic scenario, b) optimistic scenario, c) Figure 1: TLS structure arbitrary scenario The simplified TLS in Figure 1 comprises fourteen nodes, In the example in Figure 2, different degradation levels are i.e., the TLS elements (nodes S1 and S2 are sources, i.e., shown. The degradation level 1 reflects the states with a suppliers; node N1 – Main Warehouse which receives the failure in a single element that does not result in any other products from the suppliers; nodes N2 – N6 – Regional consequently disrupted TLS elements. The advantage of Warehouses who receives the products from the main using the robustness computation by the genome method is Warehouses; node C1 – Customers region which is served that this allows both disruption scenario identification and by the main warehouses; nodes C2 – C6 – Customers regions the corresponding path of the ripple effect. As such, the which are served by the regional warehouses) and thirteen results of this structural analysis can be used further to arcs. optimize the network reconfiguration paths with The computational example for the TLS design given in consideration of the operational TLS parameters such as Figure 1 is considered. Based on the genome method, the capacities, processing intensities, and inventory storage. edges 1, 2, and 3 have been shown to be critical in the TLS However, even in the structural analysis without a considered. In Figure 2, the corresponding robustness parametric optimization, the method proposed allows the assessments and disruption scenarios are presented critical TLS elements, the disruption of which would result according to different structural degradation levels. in a non-fulfillment state, to be identified. In Figure 2, the structure dynamics scenarios are depicted. 5 Conclusions Si ,i ,..,i denotes the structural states where disrupted 1 2 k The aim of this research was to establish an explicit operations (edges) in the TLS from Figure 1 are described interrelation between the disruption scenario recognition by indexes i1 , i2 , ..., ik on the abscissa scale. The state and the optimization of the TLS reconfiguration paths – a transitions are disruption-driven. In this context, a state distinctive and substantial contribution made by our study. 89 Our study explicitly includes the risk aversion of decision- European Journal of Operational Research, 224(2), makers both in the disruption scenario detection and pp. 313-323. reconfiguration path optimization. Such a combination is [Iva13a] Ivanov, D., B. Sokolov, A. Pavlov (2013a). Dual unique in the literature and mimics the complexity of problem formulation and its application to optimal business reality affording for more realistic applications to redesign of an integrated production–distribution TLS design and sourcing planning. A distinctive feature and network with structure dynamics and ripple effect novelty of the proposed approach is that on a single considerations. 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