4. Multigrammatical representation of local sociotechnological systems and criterion of their resilience

Let us consider first the local case, where all humans live and work at a single place. If so, decomposition of the human socium, having placed at this location, may begin from the unitary rule

$$\text{1-score} \to \mathbf{1} \cdot \text{structures, } \mathbf{1} \cdot \text{persons},\tag{28}$$

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

where non-terminal object structures is a start point for all business and state structures, while non-terminal object persons is similarly a start point for individuals, not entering any of the aforementioned structures.

Object structures is the head of the single unitary rule

$$\text{structutes} \to m\_1 \cdot \text{str}\_1, \dots, m\_k \cdot \text{str}\_k,\tag{29}$$

that means there are m<sup>1</sup> structures (of type) str1, …, mk structures (of type) strk; if any str i of str1, …, strk is unique, then mi ¼ 1.

Any structure may be decomposed to substructures, individual positions, and multiple access technological systems (MATS), used by personnel of this structure and its substructures. Relevant unitary rules would have the following form:

$$str \to n\_1 \cdot pstn\_1, n\_p \cdot pstn\_p, m\_1 \cdot str\_1, \dots, m\_s \cdot str\_s,\tag{30}$$
 
$$l\_1 \cdot tech\_1, \dots, l\_t \cdot tech\_t,$$

which means there are n1, …, np positions pstn1, …, pstnp and m1, …, ms substructures str1, …, strs, as well as l1, …, lt MATS tech1, …, techt (all, respectively). Every substructure is decomposed in the same way recursively until substructures, which multigrammatical representation is like

$$str \rightarrow n\_1 \cdot pstn\_1, \ldots, n\_p \cdot pstn\_p, l\_1 \cdot techn\_1, \ldots, l\_t \cdot techn\_t \tag{31}$$

or

If this DIS is affected by NHI Z ¼ f g z<sup>3</sup> , then

Natural Hazards - Risk, Exposure, Response, and Resilience

tion of the order, i.e., manufacturing of two aircrafts. ∎

system with resource base, reduced by this NHI.

basic notions, which will be used below.

¼ f3 � fuselage=z1; 50 � kW=z1; 4 � frame=z2; 5 � engine=z2; 8 � wheel=z2; 150 � kW=z2g,

and, as may be seen without generation, DIS is vulnerable to this NHI while order q completion. This means that destruction of petrol storage, necessary for refueling of transportation vehicle, which, in turn, is necessary for assembled wing removal to the place of the final assembling of aircraft, makes impossible comple-

This example is a primary illustration of multigrammatical representation and modeling of chain effects, occurring in distributed industrial systems as a result of

Now, we have the widest criterion of resilience of distributed industrial system, completing single order, to natural hazard impact. The thing is that in general case

It is evident that DSTS would be considered resilient to NHI, if the aforementioned flow would be completed by the producing (industrial) segment of this

Before we move to further discourse, let us clarify interconnections between

manufacturing process and produce resources, which, in turn, are necessary for their own existence and operation, as well as for all other humans and devices, not participating in the manufacturing process and thus entering only consuming

The described decomposition of STS will be exclusively important while studying issues, concerning consequences of total robotization of the industry, logistics, and various services that lead to massive unemployment, and the main problem to solve this will be to assess, whether global technosphere and natural resource base would be able to provide sufficient quality of life of unemployed people, as well as

However, here we shall use the described decomposition of STS for continuation of development of criterial base of their resilience. To consider distributed STS at

4. Multigrammatical representation of local sociotechnological systems

Let us consider first the local case, where all humans live and work at a single place. If so, decomposition of the human socium, having placed at this location,

socium ! 1 � structures, 1 � persons, (28)

As it was said in Section 1, any sociotechnological system includes anthropogenic and technogenic parts—humans and used by them technical devices (systems). We call them human and technological segments (STS HS and STS TS, respectively). From the order side, STS include producing (industrial) and consuming segments (STS IS and STS CS, respectively), both consisting of humans

there is a flow of such orders, generated by human segment of distributed

and devices. So, there are humans and devices that participate in the

other groups of population, being out of the producing segment.

all, we shall begin from the simplest case of local STS.

and criterion of their resilience

may begin from the unitary rule

v � Δv Zð Þ

NHI.

segment.

248

sociotechnological system.

$$str \rightarrow n\_1 \cdot pstn\_1, \ldots, n\_p \cdot pstn\_{p'} \tag{32}$$

i.e., they have no any substructures, but in general case may have MATS, providing their operation.

MATS, in turn, operate due to some attached (affiliated) personnel, which mission is to maintain technological system in the active state and apply it according to its destination. Also, MATS may consist of some subsystems, each with its own personnel, and its multigrammatical representation in general case may be as follows:

$$\text{tech}\rightarrow n\_1 \cdot p \text{st} n\_1, \dots, n\_p \cdot p \text{st} n\_p, l\_1 \cdot \text{tech}\_1, \dots, l\_s \cdot \text{tech}\_s,\tag{33}$$

$$
\mathcal{C} \textit{tech} \longrightarrow l\_1 \cdot \textit{tech}\_1, \ldots, l\_s \cdot \textit{tech}\_s,\tag{34}
$$

the latter case corresponding to the fully robotized (unmanned) system. Every techi, in turn, may be decomposed recursively until terminal objects, which names have been placed only in the bodies of unitary rules.

Concerning the second multiobject from the body of UR (28), it may be approved that all set of individuals of the considered STS may be divided to subsets (classes), each joining person with the similar sets of personal technical devices and consumed resources. This may be represented by unitary rule

$$person \rightarrow n\_1 \cdot person\_1, \ldots, n\_l \cdot person\_l,\tag{35}$$

and

$$person\_i \to k\_1^i \cdot res\_1^i, \dots, k\_{r\_i}^i \cdot res\_{r\_i}^i, m\_1^i \cdot dev\_1^i, \dots, m\_{l\_i}^i \cdot dev\_{l\_i}^i,\tag{36}$$

that means each person, belonging to the ith class, during the predefined period of time consumes k<sup>i</sup> <sup>1</sup>, …, k<sup>i</sup> <sup>r</sup> units of resources res<sup>i</sup> <sup>1</sup>, …, res<sup>i</sup> ri and is using m<sup>i</sup> <sup>1</sup>, …, m<sup>i</sup> li devices devi <sup>1</sup>, …, dev<sup>i</sup> li , respectively.

From here, it is evident that the same assignment of the consumed resources and used devices must be done regarding all positions, having placed in structures, described by UR (30)–(32). Relevant unitary rules are similar to (36):

$$p\text{ }p\text{st}\\
n \rightarrow k\_1 \cdot r\text{rs}\_1, \dots, k\_r \cdot r\text{rs}\_r, m\_1 \cdot dev\_1, \dots, m\_l \cdot dev\_l,\tag{37}$$

or

$$p\text{st}n \to k\_1 \cdot res\_1, \dots, k\_r \cdot res\_r \tag{38}$$

where MS q = {n<sup>0</sup>

for STS.

NHI Δv.

251

<sup>1</sup> � or1, …, n<sup>0</sup>

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

However, there is no any duplication.

their bodies, do not enter STS IS.

where S ¼ , socium, RH ∪ RI . :∎

base v is resilient to NHI Δv, if

the following set of unitary rules RH:

completed by STS during the considered time period. Of course, set R would contain unitary rules, representing STS IS capabilities to complete TEO.

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

Before we shall formulate following statements, let us clarify one important issue, concerning representation of resources, consumed by producing MATS and devices, entering industrial segment of STS. Namely, if MATS/device enter STS IS, it seems that its resource consumption is accounted twice—in RH as well as in RI.

RH contains representation of resources, contained by producing MATS/device during all considered time period independently of amounts of produced by OR. Most often it may be electrical power, consumed for MATS/device maintenance in the state, ready for operation, which is represented by MO n � kW (number of consumed kW). In general case, this resource is "readiness" of MATS/device to work, represented by MO 1�ready-tech or 1�ready-dev. Of course, the same MO must present in the resource base of STS. NHI may eliminate such OR from RB that reflects transfer of MATS/device out of operation, so RB becomes insufficient

At the same time, URs, representing MATS/device productive capabilities and having placed in the set RI, describe resources, consumed while STS produces OR and necessary, namely, for this operation cycle. Obviously, amounts of resources,

As seen from the said, there is no any double count, and both parts of consumed

To unify and to distinguish representation of producing MATS/devices, we shall include the body of any such unitary rule with head x multiobject 1�ready-x. Thus, all other MATS/devices, entering set RH and represented by UR without such MO in

Now, we may formulate a primary criterion of sufficiency of the resource base of STS during the considered time period. Let TMS v be the resource base of STS at the beginning of this period, while unitary multigrammars SH ¼ , socium, RH . and SI ¼ , tb, RI . represent human and industrial segments of this STS.

v⊆v, (43)

, (44)

∃ v ∈ VS

sufficiency of this RB may be recognized according to (25), if to suppose

If v contains not only terminal but also non-terminal (produced) OR, then

q ¼ f g 1 � socium . Not more difficult is generalized criterion of STS sustainability to

∃ v; v<sup>0</sup> h i ∈ VSqð Þ <sup>v</sup>�Δ<sup>v</sup> <sup>v</sup>⊆v<sup>0</sup>

where q = {1 socium}. Otherwise, this STS is vulnerable to this NHI. ∎

Statement 8. STS, represented by UMG S ¼ , socium, RH ∪ RI . , with resource

Example 6. Let sociotechnical system contain human segment, represented by

socium ! 1 � structures, 1 � persons,

structures ! 1 � office, 1 � food–factory, 1 � generation–facility,

consumed while OR production, depend on amounts of produced OR.

Statement 7. Resource base v is sufficient to STS, if

collections of OR are summarized, when their total amounts are obtained.

<sup>m</sup> � orm} is total external order (TEO), which would be

(the latter retains possibility of "deviceless" positions). All devices, represented by multiobjects, having placed in the body of UR (37), are in private use of a person; holding this position, for all time this person is assigned to this position (i.e., these devices are not of multiple access and are not the property of the person).

Let us take into account that every MATS, as well as every device, used by the person also consumes resources, necessary for its operation. To represent this obstacle, it is sufficient to use URs like

$$theh \to k\_1 \cdot res\_1, \dots, k\_l \cdot res\_l,\tag{39}$$

regarding "terminal" MATS and subsystems of "non-terminal" MATS, which are not decomposed during STS description. Similar URs define resources, consumed by devices:

$$dev \to k\_1 \cdot res\_1, \ldots, k\_d \cdot res\_d. \tag{40}$$

Let us denote SH unitary multiset grammar, which title object is socium, and scheme RH contains all unitary rules, representing considered human segment of STS. By this it is evident that total amount of resources, consumed by this segment during predefined time interval, is VSH and, namely, this amount must be produced by the STS industrial segment for STS operation. Since then it is obvious that interconnection and intersection between human and technological segments are formed by URs, defining STS IS:

$$res \longrightarrow k\_1 \cdot res\_1, \ldots, k\_m \cdot res\_m,\tag{41}$$

which means STS IS manufactures one unit of resource resi, consuming during production cycle k1, …, km units of resources res1, …, resm, respectively.

As may be seen, industrial segments of considered STS do not produce nothing but OR, necessary for the existence of humans of this STS, and structures, having placed in (29)–(32), are also producing nothing. By this reason any such STS is closed not only in the sense it has no contact with external systems, which may supply it by resources, but also in the sense that it does not produce any OR for mentioned external systems, i.e., does not complete any orders of such systems.

However, it is not difficult to represent STS, which do complete orders of external systems: it is sufficient to join to the body of UR (28) multiobject 1 � order and to include to the set RH of unitary rules, representing human segment of STS, UR

$$
overline{r} \to n\_1' \cdot or\_1, \dots, n\_m' \cdot or\_m \tag{42}$$

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

where MS q = {n<sup>0</sup> <sup>1</sup> � or1, …, n<sup>0</sup> <sup>m</sup> � orm} is total external order (TEO), which would be completed by STS during the considered time period. Of course, set R would contain unitary rules, representing STS IS capabilities to complete TEO.

Before we shall formulate following statements, let us clarify one important issue, concerning representation of resources, consumed by producing MATS and devices, entering industrial segment of STS. Namely, if MATS/device enter STS IS, it seems that its resource consumption is accounted twice—in RH as well as in RI.

However, there is no any duplication.

that means each person, belonging to the ith class, during the predefined period

From here, it is evident that the same assignment of the consumed resources and

(the latter retains possibility of "deviceless" positions). All devices, represented by multiobjects, having placed in the body of UR (37), are in private use of a person; holding this position, for all time this person is assigned to this position (i.e., these

Let us take into account that every MATS, as well as every device, used by the

regarding "terminal" MATS and subsystems of "non-terminal" MATS, which are not decomposed during STS description. Similar URs define resources, con-

Let us denote SH unitary multiset grammar, which title object is socium, and scheme RH contains all unitary rules, representing considered human segment of STS. By this it is evident that total amount of resources, consumed by this segment during predefined time interval, is VSH and, namely, this amount must be produced by the STS industrial segment for STS operation. Since then it is obvious that interconnection and intersection between human and technological segments are

which means STS IS manufactures one unit of resource resi, consuming during

As may be seen, industrial segments of considered STS do not produce nothing but OR, necessary for the existence of humans of this STS, and structures, having placed in (29)–(32), are also producing nothing. By this reason any such STS is closed not only in the sense it has no contact with external systems, which may supply it by resources, but also in the sense that it does not produce any OR for mentioned external systems, i.e., does not complete any orders of such systems. However, it is not difficult to represent STS, which do complete orders of external systems: it is sufficient to join to the body of UR (28) multiobject

1 � order and to include to the set RH of unitary rules, representing human segment

<sup>1</sup> � or1, …, n<sup>0</sup>

production cycle k1, …, km units of resources res1, …, resm, respectively.

order ! n<sup>0</sup>

devices are not of multiple access and are not the property of the person).

person also consumes resources, necessary for its operation. To represent this

<sup>1</sup>, …, res<sup>i</sup>

pstn ! k<sup>1</sup> � res1, …, kr � resr, m<sup>1</sup> � dev1, …, ml � devl, (37)

pstn ! k<sup>1</sup> � res1, …, kr � resr (38)

tech ! k<sup>1</sup> � res1, …, kt � rest, (39)

dev ! k<sup>1</sup> � res1, …, kd � resd: (40)

res ! k<sup>1</sup> � res1, …, km � resm, (41)

<sup>m</sup> � orm, (42)

ri and is using m<sup>i</sup>

<sup>1</sup>, …, m<sup>i</sup> li

<sup>r</sup> units of resources res<sup>i</sup>

described by UR (30)–(32). Relevant unitary rules are similar to (36):

used devices must be done regarding all positions, having placed in structures,

of time consumes k<sup>i</sup>

<sup>1</sup>, …, dev<sup>i</sup> li

obstacle, it is sufficient to use URs like

formed by URs, defining STS IS:

devices devi

or

sumed by devices:

of STS, UR

250

<sup>1</sup>, …, k<sup>i</sup>

, respectively.

Natural Hazards - Risk, Exposure, Response, and Resilience

RH contains representation of resources, contained by producing MATS/device during all considered time period independently of amounts of produced by OR. Most often it may be electrical power, consumed for MATS/device maintenance in the state, ready for operation, which is represented by MO n � kW (number of consumed kW). In general case, this resource is "readiness" of MATS/device to work, represented by MO 1�ready-tech or 1�ready-dev. Of course, the same MO must present in the resource base of STS. NHI may eliminate such OR from RB that reflects transfer of MATS/device out of operation, so RB becomes insufficient for STS.

At the same time, URs, representing MATS/device productive capabilities and having placed in the set RI, describe resources, consumed while STS produces OR and necessary, namely, for this operation cycle. Obviously, amounts of resources, consumed while OR production, depend on amounts of produced OR.

As seen from the said, there is no any double count, and both parts of consumed collections of OR are summarized, when their total amounts are obtained.

To unify and to distinguish representation of producing MATS/devices, we shall include the body of any such unitary rule with head x multiobject 1�ready-x. Thus, all other MATS/devices, entering set RH and represented by UR without such MO in their bodies, do not enter STS IS.

Now, we may formulate a primary criterion of sufficiency of the resource base of STS during the considered time period. Let TMS v be the resource base of STS at the beginning of this period, while unitary multigrammars SH ¼ , socium, RH . and SI ¼ , tb, RI . represent human and industrial segments of this STS.

Statement 7. Resource base v is sufficient to STS, if

$$\left(\exists \,\overline{v} \in \overline{V}\_{\mathbb{S}}\right) \overline{v} \subseteq v,\tag{43}$$

where S ¼ , socium, RH ∪ RI . :∎

If v contains not only terminal but also non-terminal (produced) OR, then sufficiency of this RB may be recognized according to (25), if to suppose q ¼ f g 1 � socium . Not more difficult is generalized criterion of STS sustainability to NHI Δv.

Statement 8. STS, represented by UMG S ¼ , socium, RH ∪ RI . , with resource base v is resilient to NHI Δv, if

$$\left(\exists\left\langle\overline{v},\overline{v}'\right\rangle\in\overline{V}\_{S\_q(v-\Delta v)}\right)\overline{v}\subseteq\overline{v}',\tag{44}$$

where q = {1 socium}. Otherwise, this STS is vulnerable to this NHI. ∎

Example 6. Let sociotechnical system contain human segment, represented by the following set of unitary rules RH:

socium ! 1 � structures, 1 � persons,

structures ! 1 � office, 1 � food–factory, 1 � generation–facility,

office ! 1 � top–manager, 1 � department, 1 � server � unit, department ! 1 � head–dpt, 3 � manager, persons ! 50 � person, top–manager ! 1 � mob–phone, 1 � desktop, 1 � lunch, head–dpt ! 1 � mob–phone, 1 � desktop, 1 � lunch, manager ! 1 � mob–phone, 1 � desktop, 1 � lunch, person ! 1 � lunch, server � unit ! 1 � hardware, 1 � engineer, engineer ! 1 � mob–phone, 1 � desktop, 1 � lunch, food–factory ! 1 � factory–director, 1 � food–line, food–line ! 1 � food–complex, 3 � food–maker, factory–director ! 1 � mob–phone, 1 � desktop, 1 � lunch, food–maker ! 1 � mob–phone, 1 � lunch, generation–facility ! 1 � generator, 1 � engineer, engineer ! 1 � mob � phone, 1 � desktop, 1 � lunch, generator ! 1 � ready � generator, mob � phone ! 0:001 � kW, desktop ! 0:1 � kW,

maker—with a mobile phone. Every person from the food factory also consumes one lunch. All devices consume electrical energy, in which amounts are multiplicities of OR kW in the bodies of the last four URs. The amount of electrical power, consumed by food complex (5 kW), does not depend on the number of lunches it does produce and is constant for all considered time interval. The third structure is MATS power generation facility, containing a power generator and maintained by an engineer. Generator consumption is described by UR, entering set R and containing MO 1�ready-generator, reflecting readiness of a generator to operation. Let us consider industrial segment of STS, represented by the following set of

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

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

tb ! 1 � lunch,

tb ! 1 � kW: This means that technological base of STS IS produces two types of OR—lunches

lunch ! 1 � lunch–eat, 1 � lunch–drink,

kW ! 0:01 � m3–gas,

lunch–eat ! 1 � chease–cake,

lunch–eat ! 1 � sandwich,

lunch–drink ! 1 � coffee,

lunch–drink ! 1 � tea,

lunch–drink ! 1 � juice,

chease–cake ! 100 � g–bread, 5 � g–sugar, 10 � g–chease,

sandwich ! 100 � g–bread, 10 � g–butter, 50 � g–meat,

tea ! 200 � g–water, 5 � g–sugar, 1 � tea–cube,

coffee ! 200 � g–water, 5 � g–sugar, 1 � coffee–cube,

juice ! 200 � g–fresh–juice:

VSH ¼ ff1 � ready � generator, 1 � ready � food � complex,

0:07619 � m3–gas, 69 � lunchgg,

As may be seen, the total order is

while

253

and electrical energy. The first contains "something to eat" and "something to drink". To produce 1 kW, it is necessary to deliver to the generator 0.01 cubic meter

unitary rules RI:

of gas:

hardware ! 1 � kW,

food � complex ! 1 � ready � food � complex, 5 � kW:

As seen, STS HS contains three structures—office, power generation facility, and food factory—as well as 50 persons out of these structures. Office includes one top manager, three departments, and one MATS—server, providing office operation. Each department, in turn, consists of the head of the department and three managers. The server unit is composed of hardware and an engineer, providing its operation. Every listed position is provided with a mobile phone and desktop, and the person, holding this position, consumes lunch daily. Other structures, entering this socium, are MATS food factory, consisting of a factory director, and food line, producing food, necessary for all humans of the considered socium.

Food line, in turn, is broken down into food complex and three food makers. The factory director is provided with a mobile phone and desktop, while every food

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

maker—with a mobile phone. Every person from the food factory also consumes one lunch. All devices consume electrical energy, in which amounts are multiplicities of OR kW in the bodies of the last four URs. The amount of electrical power, consumed by food complex (5 kW), does not depend on the number of lunches it does produce and is constant for all considered time interval. The third structure is MATS power generation facility, containing a power generator and maintained by an engineer. Generator consumption is described by UR, entering set R and containing MO 1�ready-generator, reflecting readiness of a generator to operation.

Let us consider industrial segment of STS, represented by the following set of unitary rules RI:

$$\begin{array}{c} tb \to 1 \cdot lunch, \\\\ tb \to 1 \cdot kW. \end{array}$$

This means that technological base of STS IS produces two types of OR—lunches and electrical energy. The first contains "something to eat" and "something to drink". To produce 1 kW, it is necessary to deliver to the generator 0.01 cubic meter of gas:


As may be seen, the total order is

$$
\overline{V}\_{S\_H} = \{ \{ \mathbf{1} \cdot \textit{ready} - \textit{generator}, \mathbf{1} \cdot \textit{ready} - \textit{food} - \textit{complex}, \mathbf{0} \}
$$

$$
\mathbf{0}.07619 \cdot \textit{m3-gas}, \mathbf{69} \cdot \textit{lunch} \},
$$

while

office ! 1 � top–manager, 1 � department, 1 � server � unit,

Natural Hazards - Risk, Exposure, Response, and Resilience

department ! 1 � head–dpt, 3 � manager,

persons ! 50 � person, top–manager ! 1 � mob–phone, 1 � desktop, 1 � lunch,

head–dpt ! 1 � mob–phone, 1 � desktop, 1 � lunch,

manager ! 1 � mob–phone, 1 � desktop, 1 � lunch,

person ! 1 � lunch,

server � unit ! 1 � hardware, 1 � engineer,

engineer ! 1 � mob–phone, 1 � desktop, 1 � lunch,

food–factory ! 1 � factory–director, 1 � food–line,

food–line ! 1 � food–complex, 3 � food–maker,

factory–director ! 1 � mob–phone, 1 � desktop, 1 � lunch,

food–maker ! 1 � mob–phone, 1 � lunch,

generation–facility ! 1 � generator, 1 � engineer,

engineer ! 1 � mob � phone, 1 � desktop, 1 � lunch,

generator ! 1 � ready � generator,

mob � phone ! 0:001 � kW,

desktop ! 0:1 � kW,

hardware ! 1 � kW,

food � complex ! 1 � ready � food � complex, 5 � kW:

As seen, STS HS contains three structures—office, power generation facility, and food factory—as well as 50 persons out of these structures. Office includes one top manager, three departments, and one MATS—server, providing office operation. Each department, in turn, consists of the head of the department and three managers. The server unit is composed of hardware and an engineer, providing its operation. Every listed position is provided with a mobile phone and desktop, and the person, holding this position, consumes lunch daily. Other structures, entering this socium, are MATS food factory, consisting of a factory director, and food line,

Food line, in turn, is broken down into food complex and three food makers. The factory director is provided with a mobile phone and desktop, while every food

producing food, necessary for all humans of the considered socium.

252

VS ¼ ff1 � ready � generator, 1 � ready � food � complex, 0:07619 � m3–gas, 6900 � g–bread, 690 � g–sugar, 690 � g–chease, 13800 � g–water, 69 � tea–cubeg, f1 � ready � generator, 1 � ready � food � complex, 0:07619 � m3–gas, 6900 � g–bread, 345 � g–sugar, 690 � g–chease, 13800 � g–fresh–juiceg, f1 � ready � generator, 1 � ready � food � complex, 0:07619 � m3–gas, 6900 � g–bread, 690 � g–sugar, 13800 � g–chease, 13800 � g–water, 69 � coffee–cubeg, f1 � ready � generator, 1 � ready � food � complex, 0:07619 � m3–gas, 6900 � g–bread, 690 � g–butter, 3450 � g–meat, 13800 � g–water, 345 � g–sugar, 69 � tea–cubeg, f1 � ready � generator, 1 � ready � food � complex, 0:07619 � m3–gas, 6900 � g–bread, 690 � g–butter, 3450 � g–meat, 13800 � g–fresh–juiceg, f1 � ready � generator, 1 � ready � food � complex, 0:07619 � m3–gas, 6900 � g–bread, 690 � g–butter, 3450 � g–meat, 13800 � g–water, 345 � g–sugar, 69 � coffee–cubegg:

complicated issues, concerning MATS/device division to producing and nonproducing, as well as implanting associated information to unitary rules, entering set RI and representing producing capabilities of the industrial segment

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

If so, all multiobjects like n � res in URs, entering both RHand RI, would be replaced by n � res=z, where z, as higher, is the name of place (area) where n units of

Let us now define the so-called total order (TO), being multiset representation of the aforementioned flow of orders, generated by human segment of DSTS. This total order must be completed by STS IS to provide STS HS after NHI by necessary resources. After that we may apply Statement 8 to TO and UMG RG, which scheme represents technological base of STS IS, reduced by elimination of unitary rules, representing elements of STS IS, which are destroyed by NHI, and to resource base, which, similarly, is reduced by elimination of MO, representing OR, located at

We shall introduce the following definition of the aforementioned total order

because it is necessary to produce only those resources, which are consumed at locations, not destroyed by NHI. Here, VH ¼ f gv is one-element set of TMS, generated by UMG SH ¼ , socium, RH . (let us remember that all locations of OR are

On the other hand, TO would be completed by technological base, also affected (partly destroyed) by the same NHI. The result of this impact may be adequately represented by elimination from the set RI those unitary rules, in which heads contain affected locations: it is clear that if point of origination of OR is destroyed,

h i <sup>a</sup>=<sup>z</sup> ! <sup>n</sup><sup>1</sup> � <sup>a</sup>1=z1; …; nm � am=zm <sup>∈</sup> <sup>R</sup>&¬ <sup>z</sup> <sup>∈</sup> <sup>Z</sup> <sup>g</sup>:

By this it is easy to formulate criterion of sustainability of distributed sociotech-

Statement 9. DSTS, represented by UMG S ¼ , socium, RH ∪ RI . , with

Q Zð Þð Þ v Zð Þ <sup>v</sup>⊆v<sup>0</sup>

As seen, (45)–(50) fully correspond to verbal description of this criterion.

∃ , v; v<sup>0</sup> . ∈ VS<sup>0</sup>

v Zð Þ¼ <sup>n</sup> � <sup>a</sup>=<sup>z</sup> <sup>j</sup> <sup>n</sup> � <sup>a</sup>=<sup>z</sup> <sup>∈</sup> <sup>v</sup>&¬ <sup>z</sup> <sup>∈</sup> <sup>Z</sup> : (47)

Q Zð Þ¼ , n<sup>1</sup> � a1=z1, …, nm � am=zm . , (49)

Q Zð Þ <sup>¼</sup> , q, R Zð Þ <sup>∪</sup> <sup>f</sup> , <sup>q</sup> ! <sup>n</sup><sup>1</sup> � <sup>a</sup>1=z1; …; nm � am=zm . <sup>g</sup> . : (50)

(46)

, (48)

So, TB of STS IS after NHI may be defined as follows:

R Zð Þ¼fh i a=z ! n<sup>1</sup> � a1=z1; …; nm � am=zm ∣

Similarly, STS IS resource base after NHI is

nological system; generalization of (43) is evident.

Otherwise, this DSTS is vulnerable to this NHI. ∎

resource base v is resilient to NHI Z, if

Q Zð Þ¼ <sup>n</sup> � <sup>a</sup>=<sup>z</sup> <sup>j</sup> <sup>n</sup> � <sup>a</sup>=<sup>z</sup> <sup>∈</sup> <sup>v</sup> & <sup>v</sup> <sup>∈</sup> VH & <sup>¬</sup> <sup>z</sup> <sup>∈</sup> <sup>Z</sup> , (45)

of STS.

Q Zð Þ :

points).

no OR is created.

where

255

S0

resource res are (would be) located.

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

places, destroyed by NHI.

$$\begin{aligned} \text{So if } v &= \{\mathbf{1} \cdot \text{ready } - \text{generator}, \mathbf{1} \cdot \text{ready } - \text{food} - \text{complex}, \\ &\quad \mathbf{1} \cdot \mathbf{m} \mathbf{3} \cdot \text{gas}, \mathbf{10000} \cdot \mathbf{g} \cdot \text{pred}, \mathbf{1000} \cdot \mathbf{g} \cdot \text{sugar}, \mathbf{1000} \cdot \mathbf{g} \cdot \text{chease}, \\ &\quad 20000 \cdot \mathbf{g} \cdot \text{water}, \mathbf{100} \cdot \text{tea} \cdot \text{cube}, \mathbf{15000} \cdot \mathbf{g} \cdot \text{fresh} \cdot \text{join}, \\ &\quad 100 \cdot \text{coffee} \cdot \text{cube}, \mathbf{100} \cdot \text{tea} \cdot \text{cube}, \mathbf{5000} \cdot \mathbf{g} \cdot \text{met}, \mathbf{1000} \cdot \mathbf{g} \cdot \text{butter} \}, \end{aligned}$$

this resource base is sufficient for this STS.

If Δv ¼ f g 0:91 � m3–gas , then the considered STS is resilient to this impact, while in the case Δv ¼ f g 0:91 � m3–gas; 15000 � g–water , this STS is vulnerable to the impact. The same result would be, if Δv ¼ f g 1 � ready � food � complex , that means food complex is destructed by the impact.∎

Let us consider now a general case of distributed sociotechnological systems.

### 5. Resilience of distributed sociotechnological systems

We shall describe distributed STS by application of techniques, considered in Section 2 regarding distributed IS, to local STS, considered in the previous Section 4.

However, we shall minimize the number of multiobjects, extended by geospatial information, by doing this only to those MO, which represent resources. This techniques not only essentially reduces the amount of work, necessary for knowledge base creation, but also excludes the necessity of consideration of rather

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

complicated issues, concerning MATS/device division to producing and nonproducing, as well as implanting associated information to unitary rules, entering set RI and representing producing capabilities of the industrial segment of STS.

If so, all multiobjects like n � res in URs, entering both RHand RI, would be replaced by n � res=z, where z, as higher, is the name of place (area) where n units of resource res are (would be) located.

Let us now define the so-called total order (TO), being multiset representation of the aforementioned flow of orders, generated by human segment of DSTS. This total order must be completed by STS IS to provide STS HS after NHI by necessary resources. After that we may apply Statement 8 to TO and UMG RG, which scheme represents technological base of STS IS, reduced by elimination of unitary rules, representing elements of STS IS, which are destroyed by NHI, and to resource base, which, similarly, is reduced by elimination of MO, representing OR, located at places, destroyed by NHI.

We shall introduce the following definition of the aforementioned total order Q Zð Þ :

$$\mathcal{Q}(\mathbf{Z}) = \{ n \cdot a/z \mid n \cdot a/z \in \overline{v} \text{ \& } \overline{v} \in \overline{V}\_H \text{ \& } \neg(z \in \overline{Z}) \},\tag{45}$$

because it is necessary to produce only those resources, which are consumed at locations, not destroyed by NHI. Here, VH ¼ f gv is one-element set of TMS, generated by UMG SH ¼ , socium, RH . (let us remember that all locations of OR are points).

On the other hand, TO would be completed by technological base, also affected (partly destroyed) by the same NHI. The result of this impact may be adequately represented by elimination from the set RI those unitary rules, in which heads contain affected locations: it is clear that if point of origination of OR is destroyed, no OR is created.

So, TB of STS IS after NHI may be defined as follows:

$$R(Z) = \{ \langle a/z \to n\_1 \cdot a\_1/z\_1, \dots, n\_m \cdot a\_m/z\_m \rangle \mid \}\tag{46}$$

$$\langle a/z \to n\_1 \cdot a\_1/z\_1, \dots, n\_m \cdot a\_m/z\_m \rangle \in R \& \cdot (z \in \overline{Z}) \}\,. \tag{46}$$

Similarly, STS IS resource base after NHI is

$$\nu(Z) = \left\{ n \cdot a/z \mid n \cdot a/z \in \nu \& \neg \left( z \in \overline{Z} \right) \right\}.\tag{47}$$

By this it is easy to formulate criterion of sustainability of distributed sociotechnological system; generalization of (43) is evident.

Statement 9. DSTS, represented by UMG S ¼ , socium, RH ∪ RI . , with resource base v is resilient to NHI Z, if

$$\left(\exists \, \leq \overline{\nu}, \overline{\nu}' > \, \in \overline{V}\_{S\_{Q(Z)}'(v(Z))}\right) \overline{\nu} \subseteq \overline{\nu}',\tag{48}$$

where

$$Q(Z) = \begin{cases} \pi\_1 \cdot a\_1 / z\_1, \dots, \pi\_m \cdot a\_m / z\_m > \cdot \\ \end{cases} \tag{49}$$

$$\mathcal{S}'\_{Q(Z)} = \, \preccurlyeq \, R(Z) \cup \{ \, \le \, q \to \, n\_1 \cdot a\_1/z\_1, \dots, n\_m \cdot a\_m/z\_m > \} > \,. \tag{50}$$

Otherwise, this DSTS is vulnerable to this NHI. ∎

As seen, (45)–(50) fully correspond to verbal description of this criterion.

VS ¼ ff1 � ready � generator, 1 � ready � food � complex, 0:07619 � m3–gas,

0:07619 � m3–gas, 6900 � g–bread, 345 � g–sugar, 690 � g–chease,

0:07619 � m3–gas, 6900 � g–bread, 690 � g–sugar, 13800 � g–chease,

0:07619 � m3–gas, 6900 � g–bread, 690 � g–butter, 3450 � g–meat,

0:07619 � m3–gas, 6900 � g–bread, 690 � g–butter, 3450 � g–meat,

0:07619 � m3–gas, 6900 � g–bread, 690 � g–butter, 3450 � g–meat,

1 � m3–gas, 10000 � g–bread, 1000 � g–sugar, 1000 � g–chease, 20000 � g–water, 100 � tea–cube, 15000 � g–fresh–juice,

If Δv ¼ f g 0:91 � m3–gas , then the considered STS is resilient to this impact, while

Let us consider now a general case of distributed sociotechnological systems.

We shall describe distributed STS by application of techniques, considered in Section 2 regarding distributed IS, to local STS, considered in the previous

information, by doing this only to those MO, which represent resources. This techniques not only essentially reduces the amount of work, necessary for knowledge base creation, but also excludes the necessity of consideration of rather

However, we shall minimize the number of multiobjects, extended by geospatial

in the case Δv ¼ f g 0:91 � m3–gas; 15000 � g–water , this STS is vulnerable to the impact. The same result would be, if Δv ¼ f g 1 � ready � food � complex , that means

100 � coffee–cube, 100 � tea–cube, 5000 � g–meat, 1000 � g–butterg,

6900 � g–bread, 690 � g–sugar, 690 � g–chease,

f1 � ready � generator, 1 � ready � food � complex,

f1 � ready � generator, 1 � ready � food � complex,

f1 � ready � generator, 1 � ready � food � complex,

13800 � g–water, 345 � g–sugar, 69 � tea–cubeg, f1 � ready � generator, 1 � ready � food � complex,

f1 � ready � generator, 1 � ready � food � complex,

13800 � g–water, 345 � g–sugar, 69 � coffee–cubegg:

So if v ¼ f1 � ready � generator, 1 � ready � food � complex,

5. Resilience of distributed sociotechnological systems

13800 � g–water, 69 � tea–cubeg,

Natural Hazards - Risk, Exposure, Response, and Resilience

13800 � g–water, 69 � coffee–cubeg,

13800 � g–fresh–juiceg,

13800 � g–fresh–juiceg,

this resource base is sufficient for this STS.

food complex is destructed by the impact.∎

Section 4.

254

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