6. Multigrammatical representation of critical infrastructures and their interconnections

We shall consider the most important critical infrastructures, which operation is absolutely necessary to provide human segment of DSTS by all required resources and services. Until it is said otherwise, we assume that all elements of these CI are stationary.

Let us begin with electricity infrastructure (EI), containing generation facilities (power plants), transforming/distributing substations (TDS), and terminal units (TU), providing delivery of electrical energy to the consumers. All listed elements are connected by links and joined by transmission networks together into electrical grids, which all together form EI [14–16].

We shall analyze EI, beginning from terminal units. Any TU in order to deliver one unit of power to the consumer, switched to this TU, must get it from the closest TDS, connected with it by link. So, unitary rule, representing this fragment of EI, would be as follows:

$$kW/z \to n \cdot kW/z', \mathbf{1} \cdot \text{link}/z'',\tag{51}$$

deliver aforementioned resources (energy carriers, EC)—most frequently, natural gas and oil products—transferred to power plants by pipelines, forming fuel infra-

Multiset-Based Assessment of Resilience of Sociotechnological Systems to Natural Hazards

Terminal units of the pipeline, which deliver resources to consumers, are

Distributing facilities of pipelines may be represented similarly to (52):

, 1 � link=z<sup>0</sup>

, 1 � link=z<sup>0</sup>

, 1 � link=z<sup>0</sup>

<sup>m</sup> are the lines, beginning at z<sup>0</sup> and ending at z1, …, zm,

:…

which means delivered energy carrier, entering this facility, is distributed to m pipes by application of the corresponding amounts of electrical power.

As it is clear, described techniques may be applied in the case of place of origination of EC, i.e., facility, producing various oil derivatives and pipeline gas,

The same techniques may be easily applied to water supply [18–20], heating

As may be seen from this short description, different critical infrastructures contain stationary facilities, producing various resources, as well as intermediate nodes and links, delivering necessary amounts of these resources to terminal units, contacting with objects of another CI, which operation depends on the mentioned

Let us note that operation of any DSTS is based not only on stationary objects of CI but also on its logistical capabilities—first of all, on mobile component of DSTS, providing relocation of material objects. Thus, sustainability of DSTS in a great degree depends on capabilities of transportation vehicles, which remained in the active state after NHI, as well as of stationary objects of transportation infrastructure, providing motion of these vehicles, as well as of the required resources (first of all, fuels and electrical energy). Such capabilities are necessary for relocation of mentioned objects from places of their creation or storage to places of

To represent transportation capabilities of DSTS, we shall use the following

res=z ! m � way � z<sup>0</sup> � z, 1 � res=z<sup>0</sup> (57)

networks [21–23], as well as sewage networks [24]. The latter differ from all previous by direction—"generation" of sewage waters is performed by terminal points, and "delivery" is performed to the root of the network, being the outflow

where multiobject n � kW=z<sup>0</sup> represents the TU of electricity infrastructure, providing delivery of one unit of resource res from location z<sup>0</sup> to location z. This amount of energy is consumed by pump, executing resource transfer. If there are some

, 1 � res=z<sup>0</sup>

<sup>1</sup>, n<sup>1</sup> � res=z<sup>0</sup>

m, nk � res=z<sup>0</sup>

res=z ! n<sup>1</sup> � res1=z1, …, nk � resk=zk, (56)

, (54)

(55)

, where n<sup>0</sup> . 1, would be used in (54)

1,

m,

represented as heads of unitary rules of the form

DOI: http://dx.doi.org/10.5772/intechopen.83508

losses during such transfer, then MO n<sup>0</sup> � res=z<sup>0</sup>

.

<sup>1</sup>, …, z<sup>0</sup>

res=z ! n � kW=z<sup>0</sup>

res=z<sup>1</sup> ! n<sup>1</sup> � kW=z<sup>0</sup>

res=zm ! nm � kW=z<sup>0</sup>

used as fuel by power plants. This facility is described as follows:

where all multiobjects are interpreted as higher.

structure [6–9, 17].

instead of 1 � res=z<sup>0</sup>

As higher, z<sup>0</sup>

respectively.

point.

amounts.

257

their consumption.

techniques. Unitary rule

where z, z<sup>0</sup> , and z<sup>00</sup> are, respectively, locations of TU, supplying it TDS, and connecting them link. Here, z and z<sup>0</sup> may be, as usual, the points, while z<sup>00</sup> is the line, represented by coordinates of its basic points (if it is straight, two such points—start and final—are sufficient, and they are, evidently, z<sup>0</sup> and z). Value n . 1 depends, finally, on losses of power while its transfer by the link; n is a rational number (as higher in Section 4, we use multiobjects with rational multiplicities, which do not change any of definitions, introduced higher for integer case [12, 13]).

If TDS, located at point z<sup>0</sup> , is connected to terminal units, located at points z1, …, zm, this fragment of EI is represented by m unitary rules:

$$\begin{aligned} \, ^kW/z\_1 &\to n\_1 \cdot kW/z', \, \mathbf{1} \cdot \text{link}/z''\_1, \\ &\dots \\ \, ^kW/z\_m &\to n\_m \cdot kW/z', \, \mathbf{1} \cdot \text{link}/z''\_m. \end{aligned} \tag{52}$$

where z<sup>00</sup> <sup>1</sup> , …, z<sup>00</sup> <sup>m</sup> are the lines, beginning at z<sup>0</sup> and ending at z1, …, zm, respectively. Similarly, fragments of EI, consisting of connected TDS, may be described. In

this case z<sup>0</sup> would be the location of delivering substation, while z1, …, zm—the locations of substations, consuming power from it.

Thus, treelike fragment of EI is described, until z<sup>0</sup> is the location of power plant, generating electrical energy.

Power plant, in turn, may be represented by UR:

$$kW/z \to n\_1 \cdot res\_1/z\_1, \dots, n\_k \cdot res\_k/z\_k,\tag{53}$$

where n1, …, nk are the amounts of resources res1, …, resk, which must be delivered to locations z1, …, zk, respectively, in order to generate 1 kW of electrical energy at location z, from which it may be delivered by links to the closest TDS. By this, evidently, z1, …, zk are locations of terminal units of other CI, which, in turn,

Multiset-Based Assessment of Resilience of Sociotechnological Systems to Natural Hazards DOI: http://dx.doi.org/10.5772/intechopen.83508

deliver aforementioned resources (energy carriers, EC)—most frequently, natural gas and oil products—transferred to power plants by pipelines, forming fuel infrastructure [6–9, 17].

Terminal units of the pipeline, which deliver resources to consumers, are represented as heads of unitary rules of the form

$$\text{res}/\text{z} \longrightarrow n \cdot k\,\text{W}/\text{z}', \mathbf{1} \cdot \text{link}/\text{z}', \mathbf{1} \cdot \text{res}/\text{z}',\tag{54}$$

where multiobject n � kW=z<sup>0</sup> represents the TU of electricity infrastructure, providing delivery of one unit of resource res from location z<sup>0</sup> to location z. This amount of energy is consumed by pump, executing resource transfer. If there are some losses during such transfer, then MO n<sup>0</sup> � res=z<sup>0</sup> , where n<sup>0</sup> . 1, would be used in (54) instead of 1 � res=z<sup>0</sup> .

Distributing facilities of pipelines may be represented similarly to (52):

$$\begin{aligned} \left| \begin{array}{c} \text{res}/\text{z}\_{1} \rightarrow n\_{1} \cdot kW/z', \, \mathbf{1} \cdot \text{link}/\text{z}'\_{1}, n\_{1} \cdot \text{res}/\text{z}'\_{1}, \\\\ \cdots \\\\ \left| \begin{array}{c} \text{res}/\text{z}\_{m} \rightarrow n\_{m} \cdot kW/z', \, \mathbf{1} \cdot \text{link}/\text{z}'\_{m}, n\_{k} \cdot \text{res}/\text{z}'\_{m}, \end{array} \right. \end{aligned} \right| \tag{55}$$

which means delivered energy carrier, entering this facility, is distributed to m pipes by application of the corresponding amounts of electrical power. As higher, z<sup>0</sup> <sup>1</sup>, …, z<sup>0</sup> <sup>m</sup> are the lines, beginning at z<sup>0</sup> and ending at z1, …, zm, respectively.

As it is clear, described techniques may be applied in the case of place of origination of EC, i.e., facility, producing various oil derivatives and pipeline gas, used as fuel by power plants. This facility is described as follows:

$$res/z \longrightarrow n\_1 \cdot res\_1/z\_1, \ldots, n\_k \cdot res\_k/z\_k. \tag{56}$$

where all multiobjects are interpreted as higher.

The same techniques may be easily applied to water supply [18–20], heating networks [21–23], as well as sewage networks [24]. The latter differ from all previous by direction—"generation" of sewage waters is performed by terminal points, and "delivery" is performed to the root of the network, being the outflow point.

As may be seen from this short description, different critical infrastructures contain stationary facilities, producing various resources, as well as intermediate nodes and links, delivering necessary amounts of these resources to terminal units, contacting with objects of another CI, which operation depends on the mentioned amounts.

Let us note that operation of any DSTS is based not only on stationary objects of CI but also on its logistical capabilities—first of all, on mobile component of DSTS, providing relocation of material objects. Thus, sustainability of DSTS in a great degree depends on capabilities of transportation vehicles, which remained in the active state after NHI, as well as of stationary objects of transportation infrastructure, providing motion of these vehicles, as well as of the required resources (first of all, fuels and electrical energy). Such capabilities are necessary for relocation of mentioned objects from places of their creation or storage to places of their consumption.

To represent transportation capabilities of DSTS, we shall use the following techniques. Unitary rule

$$\text{res}/\text{z} \to \text{m} \cdot \text{way} - \text{z}' - \text{z}, \mathbf{1} \cdot \text{res}/\text{z}' \tag{57}$$

Now, it would be reasonable to consider in more details multigrammatical representation of the most significant elements of DSTS IS, usually named critical

We shall consider the most important critical infrastructures, which operation is absolutely necessary to provide human segment of DSTS by all required resources and services. Until it is said otherwise, we assume that all elements of these CI are

Let us begin with electricity infrastructure (EI), containing generation facilities (power plants), transforming/distributing substations (TDS), and terminal units (TU), providing delivery of electrical energy to the consumers. All listed elements are connected by links and joined by transmission networks together into electrical

We shall analyze EI, beginning from terminal units. Any TU in order to deliver one unit of power to the consumer, switched to this TU, must get it from the closest TDS, connected with it by link. So, unitary rule, representing this fragment of EI,

, and z<sup>00</sup> are, respectively, locations of TU, supplying it TDS, and

, is connected to terminal units, located at points

1 ,

m:

(52)

, 1 � link=z<sup>00</sup>

, 1 � link=z<sup>00</sup>

kW=z ! n<sup>1</sup> � res1=z1, …, nk � resk=zk, (53)

<sup>m</sup> are the lines, beginning at z<sup>0</sup> and ending at z1, …, zm, respectively.

connecting them link. Here, z and z<sup>0</sup> may be, as usual, the points, while z<sup>00</sup> is the line, represented by coordinates of its basic points (if it is straight, two such points—start and final—are sufficient, and they are, evidently, z<sup>0</sup> and z). Value n . 1 depends, finally, on losses of power while its transfer by the link; n is a rational number (as higher in Section 4, we use multiobjects with rational multiplicities, which do not

, 1 � link=z00, (51)

kW=z ! n � kW=z<sup>0</sup>

change any of definitions, introduced higher for integer case [12, 13]).

kW=z<sup>1</sup> ! n<sup>1</sup> � kW=z<sup>0</sup>

kW=zm ! nm � kW=z<sup>0</sup>

:…

Similarly, fragments of EI, consisting of connected TDS, may be described. In this case z<sup>0</sup> would be the location of delivering substation, while z1, …, zm—the

Thus, treelike fragment of EI is described, until z<sup>0</sup> is the location of power plant,

where n1, …, nk are the amounts of resources res1, …, resk, which must be delivered to locations z1, …, zk, respectively, in order to generate 1 kW of electrical energy at location z, from which it may be delivered by links to the closest TDS. By this, evidently, z1, …, zk are locations of terminal units of other CI, which, in turn,

z1, …, zm, this fragment of EI is represented by m unitary rules:

locations of substations, consuming power from it.

Power plant, in turn, may be represented by UR:

6. Multigrammatical representation of critical infrastructures and

infrastructures.

stationary.

would be as follows:

where z, z<sup>0</sup>

where z<sup>00</sup>

256

If TDS, located at point z<sup>0</sup>

<sup>1</sup> , …, z<sup>00</sup>

generating electrical energy.

their interconnections

Natural Hazards - Risk, Exposure, Response, and Resilience

grids, which all together form EI [14–16].

means that one unit of resource res may be removed from the place of its storage z<sup>0</sup> to the place of its consumption z by any of ways, which are available by mobile component of technological segment of DSTS. It is important that multiplicity m is the mass of one unit of resource res, measured in some fixed for DSTS units (e.g., kg). According to the techniques of multigrammatical representation of similar problems, proposed in [12, 13], OR way � z<sup>0</sup> � z is detailed by unitary rules like

$$
\mathbf{zway} - \mathbf{z}' - \mathbf{z} \to \mathbf{1} \cdot \mathbf{z}\_1,\\
l\_1 \cdot \mathbf{e}/\mathbf{z}',\tag{58}
$$

$$z\_1 \to \mathbf{1} \cdot z\_2, l\_2 \cdot e/z',\tag{59}$$

e=z<sup>0</sup> ! 1 � vel=z<sup>0</sup>

Multiset-Based Assessment of Resilience of Sociotechnological Systems to Natural Hazards

If petrol-moved ground transport is used, then (65) becomes

DOI: http://dx.doi.org/10.5772/intechopen.83508

structure to transportation infrastructure is represented.

DSTS.

industrial systems in [2].

may be formulated as follows.

the remained resource base.

dual to UMG S.

259

the following multisets:

2. n<sup>1</sup> � pstn1; …; nk � pstnk

e=z<sup>0</sup> ! 1 � vel=z<sup>0</sup>

, k � kW=z<sup>0</sup>

, k � l � petrol=z<sup>0</sup>

and electricity infrastructure is connected by terminal unit, having placed at z<sup>0</sup>

where multiobject k � l � petrol=z<sup>0</sup> represents the amount of liters of petrol, required for relocation of one kg-km by vehicle vel. Thus, connection of fuel infra-

detailing may be done for every concrete vehicle, not only a class of vehicles.

The same description may be used for aircrafts, helicopters, ships, etc., and such

Possibility of non-terminal multiobjects in the resource base of DSTS provides opportunity of representation of such ways of resource relocation, which use different vehicles, moving over one and the same path, and even different vehicles, moving over sequential fragments of the path. Such techniques will be considered in the separate publication, as well as issues, concerning recovery of the vulnerable

Some primary results on the assessment of capabilities of vulnerable DSTS are presented in the next section; these results are based on the approach, applied to

Problem, which is considered in this section, is reverse to the previous one and

Let DSTS be vulnerable in the sense of criterion, formulated by Statement 9, i.e., its producing segment and resource base, affected by NHI, are not sufficient for

Question is that what maximal part (subsystem) of DSTS may stay active, being

Solution of this problem, proposed in [2], is based on application of the so-called dual multiset grammars for generation of orders, which may be completed given

Let us consider at first local case, which in the simplest form may be described by UMG S ¼ , socium, RH ∪ RI . , resource base v, and NHI Δv, which in aggregate

We shall use MG <sup>S</sup>�<sup>1</sup> <sup>¼</sup> , <sup>v</sup> � <sup>Δ</sup>v, R�<sup>1</sup> . , where <sup>R</sup> <sup>¼</sup> RH <sup>∪</sup> RI, which is called

1. n<sup>1</sup> � str1; …; nl f g � strl , representing integral structures, which may be active after NHI, because they have sufficient amounts of resources for operation

 , representing separate positions, entering some structures, which as a whole do not enter the previous set by the reason some

of their positions cannot be supplied by all necessary resources

As may be seen, every terminal multiset v ∈ VS�<sup>1</sup> in general case may be a join of

manufacturing facilities and resources. Similar question was for the first time posed in [2], where its objective was to get part of the order, which may be completed by

7. Assessment of maximal acting subsystem of vulnerable DSTS

completion of total order, generated by human segment of DSTS.

do not satisfy generalized criterion, represented by Statement 8.

the affected industrial system and its resource base.

supplied by sufficient amounts of resources, produced by the remained

, (65)

, (66)

.

$$z\_{k-1} \to \mathbf{1} \cdot z\_k,\\ l\_k \cdot e/z',\tag{60}$$

$$z\_k \to \mathbf{1} \cdot z', l\_{k+1} \cdot \mathbf{e}/z',\tag{61}$$

which describe path from z<sup>0</sup> to z, passing through points z1, …, zk, such that distance from z<sup>0</sup> to zk is lkþ<sup>1</sup> km; from zk to zk�<sup>1</sup> — lk km, …; from z<sup>2</sup> to z<sup>1</sup> — l<sup>2</sup> km; and from z<sup>1</sup> to z — l<sup>1</sup> km. As becomes evident, application of unitary rules (57)–(61) provides generation of multiset:

⋯

$$\{\mathbf{1}\cdot\mathbf{res}/\mathbf{z}^{\prime}, \mathbf{m}\cdot\mathbf{z}^{\prime}, \mathbf{K}\cdot\mathbf{e}/\mathbf{z}^{\prime}\},\tag{62}$$

where

$$K = m \cdot \sum\_{i=1}^{k+1} l\_k \tag{63}$$

is the number of kg-km, which must be removed from point z<sup>0</sup> to point z by the aforementioned mobile segment of DSTS in order to relocate one unit of resource res from z<sup>0</sup> to z. If the total order contains multiobject M � res=z, then it is necessary to remove from z<sup>0</sup> to z M � K kg km. So if the resource base of DSTS, no matter, before NHI or after it, contains such or more amount of kg-km, this operation is possible; otherwise, it is not.

As seen, the presence of multiobjects like K � e=z<sup>0</sup> in the RB describes the capability of mobile segment of DSTS to relocate resources between its points, no matter what kind of transport is used (trains, trucks, aircrafts, helicopters, ships, etc.). NHI may eliminate some part of such resource, thus reducing transportation capabilities of DSTS. Also, if NHI strikes some points, entering path from z<sup>0</sup> to z, corresponding URs will be extracted from scheme R. So, NHI may destroy transportation segment of DSTS both in topological and resource dimensions.

Of course, there may be different ways of one and the same resource relocation. Representation of any of them begins from UR like (58), which the head is way � z<sup>0</sup> � z.

One more issue to be considered here is interconnection of the transportation infrastructure with other CI (first of all, electricity and fuel). This one may be done by including to scheme R unitary rules like

$$\text{let}/\text{z}' \rightarrow \mathbf{1} \cdot \text{vel}/\text{z}', \newline k \cdot \text{res}-mov-\text{vel}/\text{z}',\tag{64}$$

which means relocation of one kg km from place z<sup>0</sup> may be done by vehicle vel and this operation requires k units of resource, used by this vehicle for motion. If electricity-moved ground transport is used, then (64) becomes

Multiset-Based Assessment of Resilience of Sociotechnological Systems to Natural Hazards DOI: http://dx.doi.org/10.5772/intechopen.83508

$$\text{Re}/\text{z}' \rightarrow \mathbf{1} \cdot \text{vel}/\text{z}', \mathbf{k} \cdot \text{kW}/\text{z}',\tag{65}$$

and electricity infrastructure is connected by terminal unit, having placed at z<sup>0</sup> . If petrol-moved ground transport is used, then (65) becomes

$$\text{let}/\text{z}' \rightarrow \mathbf{1} \cdot \text{vel}/\text{z}', \, k \cdot \text{l} - \text{pertol}/\text{z}',\tag{66}$$

where multiobject k � l � petrol=z<sup>0</sup> represents the amount of liters of petrol, required for relocation of one kg-km by vehicle vel. Thus, connection of fuel infrastructure to transportation infrastructure is represented.

The same description may be used for aircrafts, helicopters, ships, etc., and such detailing may be done for every concrete vehicle, not only a class of vehicles.

Possibility of non-terminal multiobjects in the resource base of DSTS provides opportunity of representation of such ways of resource relocation, which use different vehicles, moving over one and the same path, and even different vehicles, moving over sequential fragments of the path. Such techniques will be considered in the separate publication, as well as issues, concerning recovery of the vulnerable DSTS.

Some primary results on the assessment of capabilities of vulnerable DSTS are presented in the next section; these results are based on the approach, applied to industrial systems in [2].

### 7. Assessment of maximal acting subsystem of vulnerable DSTS

Problem, which is considered in this section, is reverse to the previous one and may be formulated as follows.

Let DSTS be vulnerable in the sense of criterion, formulated by Statement 9, i.e., its producing segment and resource base, affected by NHI, are not sufficient for completion of total order, generated by human segment of DSTS.

Question is that what maximal part (subsystem) of DSTS may stay active, being supplied by sufficient amounts of resources, produced by the remained manufacturing facilities and resources. Similar question was for the first time posed in [2], where its objective was to get part of the order, which may be completed by the affected industrial system and its resource base.

Solution of this problem, proposed in [2], is based on application of the so-called dual multiset grammars for generation of orders, which may be completed given the remained resource base.

Let us consider at first local case, which in the simplest form may be described by UMG S ¼ , socium, RH ∪ RI . , resource base v, and NHI Δv, which in aggregate do not satisfy generalized criterion, represented by Statement 8.

We shall use MG <sup>S</sup>�<sup>1</sup> <sup>¼</sup> , <sup>v</sup> � <sup>Δ</sup>v, R�<sup>1</sup> . , where <sup>R</sup> <sup>¼</sup> RH <sup>∪</sup> RI, which is called dual to UMG S.

As may be seen, every terminal multiset v ∈ VS�<sup>1</sup> in general case may be a join of the following multisets:


means that one unit of resource res may be removed from the place of its storage z<sup>0</sup> to the place of its consumption z by any of ways, which are available by mobile component of technological segment of DSTS. It is important that multiplicity m is the mass of one unit of resource res, measured in some fixed for DSTS units (e.g., kg). According to the techniques of multigrammatical representation of similar problems, proposed in [12, 13], OR way � z<sup>0</sup> � z is detailed by unitary

Natural Hazards - Risk, Exposure, Response, and Resilience

way � z<sup>0</sup> � z ! 1 � z1, l<sup>1</sup> � e=z<sup>0</sup>

z<sup>1</sup> ! 1 � z2, l<sup>2</sup> � e=z<sup>0</sup>

⋯ zk�<sup>1</sup> ! 1 � zk, lk � e=z<sup>0</sup>

which describe path from z<sup>0</sup> to z, passing through points z1, …, zk, such that distance from z<sup>0</sup> to zk is lkþ<sup>1</sup> km; from zk to zk�<sup>1</sup> — lk km, …; from z<sup>2</sup> to z<sup>1</sup> — l<sup>2</sup> km; and from z<sup>1</sup> to z — l<sup>1</sup> km. As becomes evident, application of unitary rules (57)–(61)

; m � z<sup>0</sup>

K ¼ m � ∑

kþ1 i¼1

is the number of kg-km, which must be removed from point z<sup>0</sup> to point z by the aforementioned mobile segment of DSTS in order to relocate one unit of resource res from z<sup>0</sup> to z. If the total order contains multiobject M � res=z, then it is necessary to remove from z<sup>0</sup> to z M � K kg km. So if the resource base of DSTS, no matter, before NHI or after it, contains such or more amount of kg-km, this operation is

As seen, the presence of multiobjects like K � e=z<sup>0</sup> in the RB describes the capability of mobile segment of DSTS to relocate resources between its points, no matter what kind of transport is used (trains, trucks, aircrafts, helicopters, ships, etc.). NHI may eliminate some part of such resource, thus reducing transportation capabilities of DSTS. Also, if NHI strikes some points, entering path from z<sup>0</sup> to z, corresponding URs will be extracted from scheme R. So, NHI may destroy transportation segment

Of course, there may be different ways of one and the same resource relocation.

One more issue to be considered here is interconnection of the transportation infrastructure with other CI (first of all, electricity and fuel). This one may be done

which means relocation of one kg km from place z<sup>0</sup> may be done by vehicle vel and this operation requires k units of resource, used by this vehicle for motion. If

, k � res � mov � vel=z<sup>0</sup>

Representation of any of them begins from UR like (58), which the head is

, lkþ<sup>1</sup> � e=z<sup>0</sup>

;K � e=z<sup>0</sup> f g, (62)

zk ! 1 � z<sup>0</sup>

1 � res=z<sup>0</sup>

, (58)

, (59)

, (60)

, (61)

lk (63)

, (64)

rules like

where

way � z<sup>0</sup> � z.

258

provides generation of multiset:

possible; otherwise, it is not.

of DSTS both in topological and resource dimensions.

e=z<sup>0</sup> ! 1 � vel=z<sup>0</sup>

electricity-moved ground transport is used, then (64) becomes

by including to scheme R unitary rules like


In general case

$$|\overline{V}\_{\mathbb{S}^{-1}}| \ge \mathbf{1},\tag{67}$$

Acknowledgements

DOI: http://dx.doi.org/10.5772/intechopen.83508

Author details

Igor Sheremet

261

The author is grateful to Prof. Fred Roberts and Prof. Don Saari for their useful

Financial University under the Government of Russian Federation, Moscow, Russia

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: sheremet@rfbr.ru

provided the original work is properly cited.

discussions and to Prof. Alexey Gvishiani for his permanent support.

Multiset-Based Assessment of Resilience of Sociotechnological Systems to Natural Hazards

so the only TMS, representing the final variant of distribution of the resources, remained in the RB after NHI, would be selected by application of some additional conditions. This task may be easily done by the use of filtering multigrammars (FMG); each FMG S ¼ , v0, R, F . along with kernel v<sup>0</sup> and scheme R contains filter F, joining conditions, which provide selection of terminal multisets, generated by application of rules from scheme R [12, 13].

General case of the distributed STS is not more complicated and may be easily solved by application of the introduced techniques.
