2. Assessment of resilience of industrial systems

necessary for normal operation of some devices and/or humans; such effects may lead to the destructive consequences far away from places (areas) where

By growth of complexity of DSTS and degree of their internal interconnectivity, it becomes more and more difficult to assess such consequences and, as a whole, resilience (or, reversely, vulnerability) of DSTS to various NH. Here, we shall understand DSTS resilience to NH as its property not to reduce humans' quality of life lower than some predefined level (as was said higher, it may be determined by the amounts of resources, consumed by anthropogenic part of

Well-known approaches to formal description and solution of DSTS resilience/ vulnerability problems, integrally considered in [1], are not applicable to most practical cases by the reason of only partial adequacy of representation of the main structural and functional features of DSTS, as well as by the reason of sharply increasing computational complexity of detecting algorithms on real dimensions. As it was shown in [1, 2], multiset-based approach to such assessment is one of the most suitable perspectives from both descriptional and computational points of view. The core of this approach is representation of technological base of the industrial systems (IS), producing necessary resources, by special multiset grammar

The simplest formal definition of resilience of IS, completing some order, is based on the presumption, that if RB, reduced by NHI, is, nevertheless, sufficient for this order completion by at least one possible way, then such IS is resilient to this

However, this definition and all formalizing it relations concern only industrial systems (producing segments of DSTS) and single orders, so until now criterion of DSTS resilience in multiset-based form is unknown. The main reason for this is that there is no technique for the assessment of the whole set of orders, which may generate consuming segment of DSTS. So, this chapter is dedicated to consideration of such general case. The basic presumption for all lower discourse is that DSTS after NHI has no any opportunity to contact with external systems in order to compensate loss of resources, being the result of NHI, i.e., DSTS is a "closed system" in terms of [3, 4]. Also, NHI is considered as single instant strike, which touches some finite set of places (areas), destroying all material objects located

Section 2 contains brief consideration of the previous results on IS resilience. Section 3 is dedicated to generalization of the known criterion of IS resilience on the case, when resource base of IS contains not only primary (terminal) resources but also resources, produced by IS since the start of its operation upon the initial state of RB until the moment of NHI. Section 4 is dedicated to the multigrammatical representation of local sociotechnological systems (STS) and formulation of criterion of their resilience, while Section 5—to the general case of DSTS. The current global reality makes extremely important development of a toolkit for the assessment of resilience of multiple interconnected DSTS, producing and delivering to the consumers specific types of resources (electrical energy, fuel, water, etc.). Such DSTS are addressed usually as critical infrastructures (CI), following their critically important mission for whole countries and world regions [5–11]. The basic approach of the proposed criteria application to CI is considered in Section 6. After NHI, some subsystems of vulnerable DSTS may stay in the active state ready for operation. So, the reverse problem, concerning such subsystems detection, is studied in Section 7. Possible directions of development of the proposed approach is announced in

natural hazard (NH) occurred.

Natural Hazards - Risk, Exposure, Response, and Resilience

(MG), and its resource base (RB)—by multiset (MS).

DSTS).

impact.

there.

the conclusion.

238

Let us remind that multiset is a set of multiobjects (MO) that is written as

$$\boldsymbol{v} = \{n\_1 \cdot a\_1, \ldots, n\_m \cdot a\_m\},\tag{1}$$

where v is the name of multiset and n<sup>1</sup> � a1, …, nm � am are the multiobjects, entering this MS; the integer number ni, i ¼ 1, …, m is called multiplicity of object ai, which means, that v contains n<sup>1</sup> identical objects a1, …, nm identical objects am, and for i 6¼ j ai 6¼ aj. Set

$$\beta(\nu) = \{a\_1, \dots, a\_m\} \tag{2}$$

is called basis of multiset v. Both object a and multiobject n � a are said to be entering v that is written without ambiguity as a ∈ v and n � a ∈ v. From the substantial point of view, object a and multiobject 1 � a are equivalent. In general case, multiplicities may be not only positive integers but also positive rational numbers [12, 13]. Empty set and empty multiset are denoted f g ∅ . Further in this chapter objects will be denoted also by symbol b with indices, as well as by strings of italic symbols.

The main multiset-based tool, which would be used below, is unitary multiset grammars (UMG) (we shall use also "multigrammar" as synonym of "multiset grammar") [12, 13].

UMG is a couple S ¼ , a0, R . , where a<sup>0</sup> is called title object and R is called scheme, being the set of unitary rules (UR), having the form

$$a \to n\_1 \cdot a\_1, \dots, n\_m \cdot a\_m,\tag{3}$$

where object a is called head and list n<sup>1</sup> � a1, …, nm � am—body of this UR. List is interpreted as multiset, i.e., f g n<sup>1</sup> � a1; …; nm � am .

The so-called structural and technological interpretations of unitary rules are used in the IS resilience assessment [2].

According to structural interpretation, (3) means that some material (physical) object (unit of resource) a consists of n<sup>1</sup> objects a1, …, nm objects am (to distinguish mathematical notion "object" from the physical one, we shall use below notion "object/resource," abbreviated OR).

Technological interpretation is an extension of the structural one, so that the body of UR

$$a \to n\_1 \cdot a\_1, \dots, n\_m \cdot a\_m, n'\_1 \cdot a'\_1, \dots, n'\_k \cdot a'\_k \tag{4}$$

contains structural components (usually spare parts of the produced device), which are MO n<sup>1</sup> � a1, …, nm � am, as well as resources, which are necessary for assembling (manufacturing) a from these components and are represented by MO n0 <sup>1</sup> � a<sup>0</sup> <sup>1</sup>, …, n<sup>0</sup> <sup>k</sup> � a<sup>0</sup> k.

Example 1. Let S ¼ , aircraft, R . , where R contains the following two unitary rules:

$$\begin{aligned} \textit{aircraft} &\rightarrow \mathbf{1} \cdot \textit{fuselage}, \mathbf{2} \cdot \textit{wing}, \\ \textit{wing} &\rightarrow \mathbf{1} \cdot \textit{fame}, \mathbf{1} \cdot \textit{engine}, \mathbf{4} \cdot \textit{wheel}. \end{aligned}$$

According to structural interpretation, this means that aircraft consists of fuselage and two wings. Any of the wings consists, in turn, of frame and engine, as well as four wheels, all connected to the wing frame. Let now S<sup>0</sup> ¼ , aircraft, R<sup>0</sup> . , where R<sup>0</sup> contains the following two URs:

aircraft ! <sup>1</sup> � fuselage, <sup>2</sup> � wing, <sup>10</sup> � kW, <sup>160</sup> � mbt‐asm‐aircraft, <sup>150000</sup> � USD wing ! 1 � frame, 1 � engine, 4 � wheel, 12 � kW, 240 � mnt � asm � wing, 400000 � usd:

According to the technological interpretation of UR, this means that assembling aircraft from a fuselage and two wings requires 160 min of operation of the aircraft's assembling line, 10 kW of electrical energy, as well as 150,000 dollars being the total cost of this work. Similarly, assembling one wing from the frame, engine, and four wheels requires 12 kW, 240 min of operation of the wing's assembling line, and 400,000 dollars. ∎

As seen, UMG provide easy and natural decomposition of complicated technological systems (devices) until elementary (non-decomposed) objects and resources, used in the manufacturing process.

A set of objects, having placed in the UMG S, is denoted AS, while a set of socalled terminal objects, having placed only in bodies of UR, is denoted AS. Evidently, AS ⊂ AS. Objects, entering set AS – AS, are called non-terminal. Similarly, corresponding OR also may be terminal and non-terminal.

Mathematical semantics of unitary multiset grammars is defined in such a way that UMG S ¼ , a0, R . is applied for generation of the set of multisets (SMS) VS according to the following relations:

$$V\_{(0)} = \{\{\mathbf{1} \cdot \mathbf{a}\_0\}\},\tag{5}$$

VS ¼ f g f g 1 � fuselage; 2 � frame; 2 � engine; 8 � wheel ,

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

950000 � usdgg:∎

producing devices) as scheme R of UMG:

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

but set of TMS, generated by UMG:

 

criterion of possibility of order completion.

In general case Vq

completion.

completed, if

devices.

241

<sup>0</sup> ¼ ff1 � fuselage, 2 � frame, 2 � engine, 8 � wheel, 34 � KW, 160 � mnt � asm � aircraft, 480 � mnt � asm � wing,

Returning to the considered application of UMG, i.e., description and assessment of industrial systems, we may represent technological base (TB) of IS (set of its

where tb is the title object and R is the set of unitary rules in the technological

which means goal of this order is to obtain n<sup>1</sup> OR b1, …, nl OR bl. The set of possible variants of resource amounts, necessary for order q completion, is nothing,

completion, which usually is a consequence of some redundancy of TB (however, such redundancy is the background of IS resilience, as it will be shown below). Resource base of IS may be represented by MS v ¼ f g n<sup>1</sup> � a1; …; nk � ak in such a way that n<sup>1</sup> OR a1, …, nk OR ak are available to technological base R while orders

Described representation of TB and RB makes it quite simple to formulate

∃ v ∈ Vq

of multiobject n � r in the resource base is equivalent to the possibility of n manufacturing cycles, executed by device r while current order completion.

Such RB v is called sufficient for order q completion by IS.

Statement 1. Order q to IS with technological base R and resource base v may be

For further consideration of resilience/vulnerability issues, it is useful to unify TB and RB by including to the bodies of URS in the technological interpretation of one additional multiobject 1 � r, where r is the name of the device, which provides manufacturing (assembling) OR, defined by the head of UR. By this, the presence

Described techniques integrate TB and RB in the integral resource base, which does not contradict to the reality, because multiobjects like n � r represent, in fact, technological (active) resources of IS, along with passive resources, consumed by

Note that there may be one and the same object r in different UR bodies that reflects the capability of device r to produce one and the same OR by various ways or even to produce various OR. Moreover, in general case, there may be not only multiobjects like 1 � r in the UR bodies but also l � r, where l . 1, that, in fact, allows to represent the duration of manufacturing cycle, providing

Order, completed by IS with TB S, may be represented by MS

i.e., STMS VSq (for short we shall use Vq instead of it).

S ¼ , tb, R . , (10)

q ¼ n<sup>1</sup> � b1; …; nl f g � bl , (11)

v⊆v:∎ (13)

Sq ¼ , tbq, R ∪ , tbq ! n<sup>1</sup> � b1; …; nl f g � bl . . , (12)

. 1 because of the possibility of multiple ways of order

VS

interpretation.

$$\mathcal{V}\_{(i+1)} = \mathcal{V}\_{(i)} \cup \left( \bigcup\_{v \in V\_{(i)}} \bigcup\_{r \in R} \left\{ \pi(v, r)\_0 \right\} \right), \tag{6}$$

$$\pi(v, \le a \to n\_1 \cdot a\_1, \dots, n\_m \cdot a\_m >) = \begin{cases} v - \{n \cdot a\} + n \ast \{n\_1 \cdot a\_1, \dots, n\_m \cdot a\_m\}, \\ \qquad \text{if } n \cdot a \in v \\ \{\mathcal{Q}\} \text{ otherwise} \end{cases} \tag{7}$$

$$V\_S = V\_{(\infty)} \tag{8}$$

where UR a ! n<sup>1</sup> � a1, …, nm � am for unambiguity is represented in the angle

brackets, and þ, � , ∗ are symbols of operations on multisets (addition, subtraction of multisets, and multiplication of constant on multiset, respectively) [1, 2, 12, 13].

As seen from (5) to (8), new multisets are generated by applying all unitary rules r ∈ R to SMS Vð Þ<sup>i</sup> , created on previous i steps. Every such UR a ! n<sup>1</sup> � a1, …, nm � am is applied to MS v ∈ Vð Þ<sup>i</sup> by a special function π. If v contains MO n � a, it is replaced by MS n ∗ f g n<sup>1</sup> � a1; …; nm � am and by semantics of MS addition, and after that multiplicities of the identical objects are summarized; otherwise, the result of π application is an empty set.

Described generation process is in general case infinite, and SMS VS, defined by UMG S, is its fixed point Vð Þ <sup>∞</sup> .

Terminal multiset (TMS) v ∈ VS contains only terminal objects, i.e.,

$$
\beta(v) \subseteq \overline{A}\_{\text{St}} \tag{9}
$$

and the set of terminal multisets (STMS) is denoted VS.

Further in this chapter if it will not be said the contrary, we shall consider only finitary UMG, which define finite STMS. UMG S is finitary, if these exists i such, that Vð Þ<sup>i</sup> ¼ Vð Þ <sup>i</sup>þ<sup>1</sup> , and if so, Vð Þ<sup>i</sup> ¼ VS. The problem of recognition of UMG finitarity is algorithmically decidable [12, 13].

Example 2. As may be seen, UMG S and S<sup>0</sup> from the previous example are finitary, and, according to (5)–(8),

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

$$\begin{aligned} \overline{V}\_{\overline{S}} &= \{ \{ 1 \cdot fuselage, 2 \cdot fame, 2 \cdot engine, 8 \cdot wheel \} \}, \\ \overline{V}\_{\overline{S}'} &= \{ \{ 1 \cdot fuselage, 2 \cdot fame, 2 \cdot engine, 8 \cdot wheel, 34 \cdot KW, 4 \cdot kW, 4 \cdot kW, 4 \cdot kW \} \} \\ &= \{ 160 \cdot mmt - asm - aircraft, 480 \cdot mmt - asm - wing, 4 \cdot kW \} \\ &= 950000 \cdot usd \} \}. \end{aligned}$$

Returning to the considered application of UMG, i.e., description and assessment of industrial systems, we may represent technological base (TB) of IS (set of its producing devices) as scheme R of UMG:

$$S = \le tb, R >,\tag{10}$$

where tb is the title object and R is the set of unitary rules in the technological interpretation.

Order, completed by IS with TB S, may be represented by MS

$$q = \{n\_1 \cdot b\_1, \ldots, n\_l \cdot b\_l\},\tag{11}$$

which means goal of this order is to obtain n<sup>1</sup> OR b1, …, nl OR bl. The set of possible variants of resource amounts, necessary for order q completion, is nothing, but set of TMS, generated by UMG:

$$S\_q = \, ^\leqslant tbp,\\ R \cup \{ \, ^\leqslant tbq \to n\_1 \cdot b\_1, \dots, n\_l \cdot b\_l > \} \ge , \tag{12}$$

i.e., STMS VSq (for short we shall use Vq instead of it).

In general case Vq . 1 because of the possibility of multiple ways of order completion, which usually is a consequence of some redundancy of TB (however, such redundancy is the background of IS resilience, as it will be shown below).

Resource base of IS may be represented by MS v ¼ f g n<sup>1</sup> � a1; …; nk � ak in such a way that n<sup>1</sup> OR a1, …, nk OR ak are available to technological base R while orders completion.

Described representation of TB and RB makes it quite simple to formulate criterion of possibility of order completion.

Statement 1. Order q to IS with technological base R and resource base v may be completed, if

$$(\exists \overline{v} \in \overline{V}\_q) \overline{v} \subseteq v. \mathbf{I} \tag{13}$$

Such RB v is called sufficient for order q completion by IS.

For further consideration of resilience/vulnerability issues, it is useful to unify TB and RB by including to the bodies of URS in the technological interpretation of one additional multiobject 1 � r, where r is the name of the device, which provides manufacturing (assembling) OR, defined by the head of UR. By this, the presence of multiobject n � r in the resource base is equivalent to the possibility of n manufacturing cycles, executed by device r while current order completion.

Described techniques integrate TB and RB in the integral resource base, which does not contradict to the reality, because multiobjects like n � r represent, in fact, technological (active) resources of IS, along with passive resources, consumed by devices.

Note that there may be one and the same object r in different UR bodies that reflects the capability of device r to produce one and the same OR by various ways or even to produce various OR. Moreover, in general case, there may be not only multiobjects like 1 � r in the UR bodies but also l � r, where l . 1, that, in fact, allows to represent the duration of manufacturing cycle, providing

aircraft ! <sup>1</sup> � fuselage, <sup>2</sup> � wing, <sup>10</sup> � kW, <sup>160</sup> � mbt‐asm‐aircraft, <sup>150000</sup> � USD wing ! 1 � frame, 1 � engine, 4 � wheel, 12 � kW, 240 � mnt � asm � wing, 400000 � usd:

According to the technological interpretation of UR, this means that assembling aircraft from a fuselage and two wings requires 160 min of operation of the aircraft's assembling line, 10 kW of electrical energy, as well as 150,000 dollars being the total cost of this work. Similarly, assembling one wing from the frame, engine, and four wheels requires 12 kW, 240 min of operation of the wing's assembling line, and

As seen, UMG provide easy and natural decomposition of complicated techno-

A set of objects, having placed in the UMG S, is denoted AS, while a set of socalled terminal objects, having placed only in bodies of UR, is denoted AS. Evidently, AS ⊂ AS. Objects, entering set AS – AS, are called non-terminal. Similarly,

Mathematical semantics of unitary multiset grammars is defined in such a way that UMG S ¼ , a0, R . is applied for generation of the set of multisets (SMS) VS

> ∪ <sup>r</sup> <sup>∈</sup> <sup>R</sup> <sup>π</sup>ð Þ <sup>v</sup>;<sup>r</sup> <sup>0</sup> � � � �

> > f g ∅ otherwise

Vð Þ <sup>0</sup> ¼ f g f g 1 � a<sup>0</sup> , (5)

v � f g n � a þ n ∗ f g n<sup>1</sup> � a1; …; nm � am ,

VS ¼ Vð Þ <sup>∞</sup> , (8)

βð Þv ⊆ AS, (9)

if n � a ∈ v

, (6)

(7)

logical systems (devices) until elementary (non-decomposed) objects and

resources, used in the manufacturing process.

Natural Hazards - Risk, Exposure, Response, and Resilience

according to the following relations:

πðv; , a ! n<sup>1</sup> � a1; …; nm � am . Þ ¼

application is an empty set.

UMG S, is its fixed point Vð Þ <sup>∞</sup> .

is algorithmically decidable [12, 13].

finitary, and, according to (5)–(8),

240

corresponding OR also may be terminal and non-terminal.

<sup>V</sup>ð Þ <sup>i</sup>þ<sup>1</sup> <sup>¼</sup> <sup>V</sup>ð Þ<sup>i</sup> ∪ ∪ <sup>v</sup> <sup>∈</sup> <sup>V</sup>ð Þ<sup>i</sup>

8 ><

>:

where UR a ! n<sup>1</sup> � a1, …, nm � am for unambiguity is represented in the angle brackets, and þ, � , ∗ are symbols of operations on multisets (addition, subtraction of multisets, and multiplication of constant on multiset, respectively) [1, 2, 12, 13]. As seen from (5) to (8), new multisets are generated by applying all unitary rules r ∈ R to SMS Vð Þ<sup>i</sup> , created on previous i steps. Every such UR a ! n<sup>1</sup> � a1, …, nm � am is applied to MS v ∈ Vð Þ<sup>i</sup> by a special function π. If v contains MO n � a, it is replaced by MS n ∗ f g n<sup>1</sup> � a1; …; nm � am and by semantics of MS addition, and after that multiplicities of the identical objects are summarized; otherwise, the result of π

Described generation process is in general case infinite, and SMS VS, defined by

Further in this chapter if it will not be said the contrary, we shall consider only finitary UMG, which define finite STMS. UMG S is finitary, if these exists i such, that Vð Þ<sup>i</sup> ¼ Vð Þ <sup>i</sup>þ<sup>1</sup> , and if so, Vð Þ<sup>i</sup> ¼ VS. The problem of recognition of UMG finitarity

Example 2. As may be seen, UMG S and S<sup>0</sup> from the previous example are

Terminal multiset (TMS) v ∈ VS contains only terminal objects, i.e.,

and the set of terminal multisets (STMS) is denoted VS.

400,000 dollars. ∎

creation of one unit of OR a, represented by the head of the UR. This technique is simply implemented by the use of so-called composite objects, or composites, like t-r, where "-" is the divider, r is the unique identifier of manufacturing device, and t is the time unit (second, minute, etc.), so l�t-r means that there are sufficient l time units of work of device r to produce one unit of OR a, represented by the head of the UR. Both r and t are strings in some basic alphabet, and t does not contain divider "-".

This criterion is basic for distributed industrial systems (DIS), in which facilities are located at different places (areas) and some of them may be affected by NHI. Every such impact may destroy some of the aforementioned facilities, eliminating some local parts of TB and RB, thus reducing its capabilities for

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

To represent DIS, OR, having placed in unitary rules and multisets, are extended by geospatial information in such a way that a=z, where "/" is the divider, means that OR a is located at place z. Both a and z are the strings in some basic alphabet,

a=z ! n<sup>1</sup> � a1=z1, …, nm � am=zm, (14)

v ¼ f g n<sup>1</sup> � a1=z1; …; nk � ak=zk , (15)

q ¼ n<sup>1</sup> � b1=z1; …; nl f g � bl=zl : (16)

<sup>Δ</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> , (18)

: (17)

We use names of locations instead of their usual coordinate representations (CR), supposing that there is a separate key-addressed database, containing couples , z, X . , where key z is the name of place and X is its CR in any possible form (points of perimeter, center of the circle along with its radius, etc.), most convenient for concrete location. This database provides the simplest implementation of intersection of two locations, which is the basic operation in the algorithmics of

that means OR a may be produced at location z, if there are n<sup>1</sup> OR a<sup>1</sup> at location z1, …, nm OR am at location zm. As seen, a=z, a1=z1, …, am=zm are also composites. Representation of time resource is just the same: if MO <sup>n</sup> � <sup>t</sup>‐r=<sup>z</sup> enters UR body, that means follows: to produce OR a, located at place z, device r, located at place z,

The new moment is the representation of NHI by set z of affected by it locations

For simplicity we shall limit a variety of locations having placed in (14)–(17) by

To formulate the criterion of resilience of DIS, we shall use relation z ∈ Z that

i.e., multiset of OR, affected by NHI z, because they are located at the affected

Statement 3. DIS, completing order q, is resilient to NHI z, if reduced by it RB, v � Δv Zð Þ is sufficient for this order completion. Otherwise, this DIS is vulnerable

Z ¼ z1; …; zp

points, while in (17) every zi may be an area of any form. Also, we shall use denotation Z for the set of points entering Z (it is join of sets z1, …, zp).

points. Thus, all these OR must be eliminated from the resource base, being

Let DIS has TB R and RB v, which is sufficient for order q completion.

order completion.

excepting "/", and z is the name of location.

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

assessment of resilience of any distributed systems. Since the described extension, all UR have the form

would operate for n time units.

as well as order

(in general case, areas):

means point z enters set Z. Let us define

destroyed by the impact.

to this NHI. ∎

243

Similarly, resource base would be

If resource base of IS contains multiobjects like L�t-r, that means there are L units of time of work of device r available while current order completion.

Speaking about the use of time in UR, we must take into account that time is not fully an additive resource; it is additive regarding only separate device. If to consider the whole IS, then due to parallel operation of various devices, time, spent for order completion, may be less than in the case of their sequential application. Precise modeling of IS operation is possible on the basis of the so-called temporal multiset grammars, introduced in [2], which will be considered thoroughly in the separate publications.

Example 3. Let S<sup>0</sup> ¼ , aircraft, R . be as in Example 1, order q ¼ f g r � aircraft , and IS resource base is

> v ¼ f6 � fuselage, 10 � frame, 12 � engine, 40 � wheel, 250 � kW, 800 � mnt � asm � aircraft, 2600 � mnt � asm � wing, 1000000 � usdgg:

As seen, order q may be completed with technological base R<sup>0</sup> and resource base v, which is sufficient for this order completion.

However, if

$$\begin{aligned} v &= \{\mathsf{6} \cdot \mathsf{fuselage}, 10 \cdot \mathsf{fframe}, 3 \cdot \mathsf{engine}, 12 \cdot \mathsf{wheel}, 250 \cdot \mathsf{kW}, \mathsf{t}\}, \\ &\quad 800 \cdot \mathsf{mmt} - \mathsf{asm} - \mathsf{air} \, \mathsf{rgt}, 2600 \cdot \mathsf{mmt} - \mathsf{asm} - \mathsf{wing}, \mathsf{t}\}, \\ &\quad 10000000 \cdot \mathsf{u} \, \mathsf{s} d \}, \end{aligned}$$

then order q cannot be completed, and RB v is not sufficient, because there is lack of five engines for manufacturing four aircrafts. ∎

Let us consider now IS, affected by natural hazard impact, which may be represented by multiset Δv, defining amounts of resources, eliminated by NHI from IS resource base, so the last becomes v � Δv.

Concerning passive resources, such representation is quite evident: if NHI destroys n<sup>0</sup> OR a from n, which had placed in RB before the impact, then the remained amount of these OR will be n � n<sup>0</sup> (if n , n<sup>0</sup> or n ¼ n<sup>0</sup> , all such OR will be eliminated from RB), so respective multiobject, entering v � Δv, will be n � n<sup>0</sup> ð Þ� a. In the case of active resources, <sup>n</sup><sup>0</sup> � <sup>t</sup>‐<sup>r</sup> <sup>∈</sup> <sup>Δ</sup><sup>v</sup> means that <sup>n</sup><sup>0</sup> time units of operation of ith devise r would be lost, so this device may not execute all work, which it would do while order completion, and this obstacle may be the reason for IS vulnerability. So, similar to passive resources, the result of NHI regarding active resource would be <sup>n</sup> � <sup>n</sup><sup>0</sup> ð Þ� <sup>t</sup>‐r. If <sup>n</sup><sup>0</sup> <sup>¼</sup> <sup>∞</sup>, the result of NHI would be elimination of MO <sup>n</sup> � <sup>t</sup>‐<sup>r</sup> from R; when implemented, ∞ may be replaced by some very large number N, which is greater than any possible multiplicity, ever used in TB and RB representations.

Let IS has TB R and RB v, which is sufficient for order q completion.

Statement 2. IS, completing order q, is resilient to NHI Δv, if reduced RB v � Δv is sufficient for this order completion. Otherwise, this IS is vulnerable to this NHI.∎ Multiset-Based Assessment of Resilience of Sociotechnological Systems to Natural Hazards DOI: http://dx.doi.org/10.5772/intechopen.83508

This criterion is basic for distributed industrial systems (DIS), in which facilities are located at different places (areas) and some of them may be affected by NHI. Every such impact may destroy some of the aforementioned facilities, eliminating some local parts of TB and RB, thus reducing its capabilities for order completion.

To represent DIS, OR, having placed in unitary rules and multisets, are extended by geospatial information in such a way that a=z, where "/" is the divider, means that OR a is located at place z. Both a and z are the strings in some basic alphabet, excepting "/", and z is the name of location.

We use names of locations instead of their usual coordinate representations (CR), supposing that there is a separate key-addressed database, containing couples , z, X . , where key z is the name of place and X is its CR in any possible form (points of perimeter, center of the circle along with its radius, etc.), most convenient for concrete location. This database provides the simplest implementation of intersection of two locations, which is the basic operation in the algorithmics of assessment of resilience of any distributed systems.

Since the described extension, all UR have the form

$$a/z \to n\_1 \cdot a\_1/z\_1, \dots, n\_m \cdot a\_m/z\_m,\tag{14}$$

that means OR a may be produced at location z, if there are n<sup>1</sup> OR a<sup>1</sup> at location z1, …, nm OR am at location zm. As seen, a=z, a1=z1, …, am=zm are also composites.

Representation of time resource is just the same: if MO <sup>n</sup> � <sup>t</sup>‐r=<sup>z</sup> enters UR body, that means follows: to produce OR a, located at place z, device r, located at place z, would operate for n time units.

Similarly, resource base would be

$$\boldsymbol{\nu} = \{n\_1 \cdot \boldsymbol{a}\_1/\boldsymbol{z}\_1, \dots, n\_k \cdot \boldsymbol{a}\_k/\boldsymbol{z}\_k\},\tag{15}$$

as well as order

creation of one unit of OR a, represented by the head of the UR. This technique is simply implemented by the use of so-called composite objects, or composites, like t-r, where "-" is the divider, r is the unique identifier of manufacturing device, and t is the time unit (second, minute, etc.), so l�t-r means that there are sufficient l time units of work of device r to produce one unit of OR a, represented by the head of the UR. Both r and t are strings in some basic

If resource base of IS contains multiobjects like L�t-r, that means there are L

Speaking about the use of time in UR, we must take into account that time is not fully an additive resource; it is additive regarding only separate device. If to consider the whole IS, then due to parallel operation of various devices, time, spent for order completion, may be less than in the case of their sequential application. Precise modeling of IS operation is possible on the basis of the so-called temporal multiset grammars, introduced in [2], which will be considered thoroughly in the

Example 3. Let S<sup>0</sup> ¼ , aircraft, R . be as in Example 1, order q ¼ f g r � aircraft ,

v ¼ f6 � fuselage, 10 � frame, 12 � engine, 40 � wheel, 250 � kW,

800 � mnt � asm � aircraft, 2600 � mnt � asm � wing,

As seen, order q may be completed with technological base R<sup>0</sup> and resource base

v ¼ f6 � fuselage, 10 � frame, 3 � engine, 12 � wheel, 250 � kW,

then order q cannot be completed, and RB v is not sufficient, because there is

Let us consider now IS, affected by natural hazard impact, which may be represented by multiset Δv, defining amounts of resources, eliminated by NHI from

Concerning passive resources, such representation is quite evident: if NHI destroys n<sup>0</sup> OR a from n, which had placed in RB before the impact, then the

eliminated from RB), so respective multiobject, entering v � Δv, will be n � n<sup>0</sup> ð Þ� a. In the case of active resources, <sup>n</sup><sup>0</sup> � <sup>t</sup>‐<sup>r</sup> <sup>∈</sup> <sup>Δ</sup><sup>v</sup> means that <sup>n</sup><sup>0</sup> time units of operation of ith devise r would be lost, so this device may not execute all work, which it would do while order completion, and this obstacle may be the reason for IS vulnerability. So, similar to passive resources, the result of NHI regarding active resource would be <sup>n</sup> � <sup>n</sup><sup>0</sup> ð Þ� <sup>t</sup>‐r. If <sup>n</sup><sup>0</sup> <sup>¼</sup> <sup>∞</sup>, the result of NHI would be elimination of MO <sup>n</sup> � <sup>t</sup>‐<sup>r</sup> from R; when implemented, ∞ may be replaced by some very large number N, which is greater than any possible multiplicity, ever used in TB and RB representations. Let IS has TB R and RB v, which is sufficient for order q completion.

Statement 2. IS, completing order q, is resilient to NHI Δv, if reduced RB v � Δv is sufficient for this order completion. Otherwise, this IS is vulnerable to this NHI.∎

, all such OR will be

800 � mnt � asm � aircraft, 2600 � mnt � asm � wing,

units of time of work of device r available while current order completion.

alphabet, and t does not contain divider "-".

Natural Hazards - Risk, Exposure, Response, and Resilience

1000000 � usdgg:

1000000 � usdgg,

lack of five engines for manufacturing four aircrafts. ∎

remained amount of these OR will be n � n<sup>0</sup> (if n , n<sup>0</sup> or n ¼ n<sup>0</sup>

IS resource base, so the last becomes v � Δv.

v, which is sufficient for this order completion.

separate publications.

and IS resource base is

However, if

242

$$q = \{n\_1 \cdot b\_1 / z\_1, \dots, n\_l \cdot b\_l / z\_l\}. \tag{16}$$

The new moment is the representation of NHI by set z of affected by it locations (in general case, areas):

$$Z = \{z\_1, \ldots, z\_p\}.\tag{17}$$

For simplicity we shall limit a variety of locations having placed in (14)–(17) by points, while in (17) every zi may be an area of any form. Also, we shall use denotation Z for the set of points entering Z (it is join of sets z1, …, zp).

To formulate the criterion of resilience of DIS, we shall use relation z ∈ Z that means point z enters set Z.

Let us define

$$
\Delta v(Z) = \{ n \cdot a/z \mid n \cdot a/z \in v \: \& \; z \in \overline{Z} \}, \tag{18}
$$

i.e., multiset of OR, affected by NHI z, because they are located at the affected points. Thus, all these OR must be eliminated from the resource base, being destroyed by the impact.

Let DIS has TB R and RB v, which is sufficient for order q completion.

Statement 3. DIS, completing order q, is resilient to NHI z, if reduced by it RB, v � Δv Zð Þ is sufficient for this order completion. Otherwise, this DIS is vulnerable to this NHI. ∎

Concerning affected active resources, it is reasonable to underline that NHI may destroy them up to unrecoverable state (this may be represented by inclusion to <sup>Δ</sup><sup>v</sup> MO <sup>N</sup> � <sup>t</sup>‐r=z) or, in the better case, transfer them to the unoperational, but reparable, state, that may be represented by inclusion to Δv MO <sup>n</sup><sup>0</sup> � <sup>t</sup>‐r=z, where <sup>n</sup><sup>0</sup> is less than multiplicity <sup>n</sup> of OR <sup>t</sup>‐r=<sup>z</sup> <sup>∈</sup> <sup>v</sup>.

<sup>V</sup>ð Þ <sup>i</sup>þ<sup>1</sup> <sup>¼</sup> <sup>V</sup>ð Þ<sup>i</sup> ∪ ∪ , v, <sup>v</sup><sup>0</sup> . <sup>∈</sup> <sup>V</sup>ð Þ<sup>i</sup>

, v � f g n � a , v<sup>0</sup> � f g n � a . , if n � a ∈ v & n<sup>0</sup> � a ∈ v<sup>0</sup> & n<sup>0</sup>

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

n � a ∈ v & n<sup>0</sup> ¼ 0

, v � f g n � a þ n � n<sup>0</sup> ð Þ ∗ f g n<sup>1</sup> � a1; …; nm � am , v<sup>0</sup> � n<sup>0</sup> f g � a . ,

This definition fully corresponds to the previous verbal description and is similar to (5)–(8). The mission of function φ, defined by (21), is the same as the mission of function π, defined by (7). Some comments would be done to its second alternative, namely, the case where multiset v<sup>0</sup> does not contain OR a at all (or, just the same,

general case, when n<sup>0</sup> � a ∈ v<sup>0</sup> and non-zero multiplicity n<sup>0</sup> is less than n. As seen, the result of subtraction of the empty multiset n<sup>0</sup> f g � a , where n<sup>0</sup> ¼ 0, from multiset v<sup>0</sup>

The introduced UMG RG provide formulation of the generalized criterion of IS

Let q ¼ n<sup>1</sup> � b1; …; nl f g � bl be order, R—technological base of the industrial sys-

(That is, it contains not only terminal but also non-terminal OR). Consider UMG

Statement 4. Order q to IS with technological base R and resource base v may be

. ∈ VSqð Þ<sup>v</sup> � � <sup>v</sup>⊆v<sup>0</sup>

As seen, if RB does not contain non-terminal OR, (25) and (13) are equivalent. As higher, RB, relevant to criterion 4, is called sufficient for order q completion by

Example 4. Let S<sup>0</sup> ¼ , aircraft, R<sup>0</sup> . be as in Example 1, order q ¼ f g 4 � aircraft , and resource base of the industrial system is v ¼ f6 � fuselage; 12 � wing; 300 � kW;

Sqð Þ¼ v , tbq, Rq, v . ,

∪ f g , tbq ! 4 � aircraft . ,

(Here, UR is written in the angle brackets for unambiguity.)

∃ , v; v<sup>0</sup>

IS. Evidently, v<sup>0</sup> � v is RB, remained after completion of order q.

Rq ¼ R<sup>0</sup>

800 � mnt � asm � aircraft; 1100000 � usdg:

, and this branch of (21) is just the same, as the first alternative of (7).

Rq ¼ R ∪ , tbq ! n<sup>1</sup> � b1; …; nl f g � bl . : (24)

φ v; v<sup>0</sup> ð ; , a ! n<sup>1</sup> � a1; …; nm � am . Þ ¼

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

multiplicity n<sup>0</sup> of multiobject n<sup>0</sup> � a, entering v<sup>0</sup>

tem, and v—its resource base, such that

RG Sqð Þ¼ v , tbq, Rq, v . , where

f g ∅ otherwise,

¼

unchanged v<sup>0</sup>

completed, if

As seen,

where

245

resilience.

8

>>>>>>>><

>>>>>>>>:

∪

if n � a ∈ v & n<sup>0</sup> � a ∈ v<sup>0</sup> & n<sup>0</sup> 6¼ 0 & n<sup>0</sup> , n∨

<sup>r</sup> <sup>∈</sup> <sup>R</sup> <sup>φ</sup> <sup>v</sup>; <sup>v</sup><sup>0</sup> f g ð Þ ;<sup>r</sup> � �, (20)

VS vð Þ <sup>0</sup> ¼ Vð Þ <sup>∞</sup> : (22)

βð Þv ⊆ As: (23)

, is zero); this is equivalent to a more

:∎ (25)

≥n

(21)

, is

By this we finish a short survey of known results on resilience of industrial systems. Before we move to sociotechnological systems, let us generalize the introduced criteria.

### 3. Generalized criterion of resilience of industrial systems

As seen, both introduced criteria of IS resilience operate only terminal resources, which are used by IS for production of amounts of OR, being the goal of order. By this, they trivially repeat criterion of order completeness (12) with the only replacement of the IS initial resource base by RB, reduced by NHI.

However, if to take into account that there may be some non-terminal OR, already manufactured by IS during time interval between the start of order completion and moment of NHI, it would be sensible to consider these OR during recognition of IS resilience, or, in the other words, to generalize notion of resource base, including to RB not only terminal, but also nonterminal OR.

But, evidently, this generalization makes the introduced criteria non-applicable. Let us propose correct criterion for the case of RB, containing not only terminal but also non-terminal OR.

For this purpose we propose here so-called unitary multiset grammars with reduced generation (UMG RG).

UMG RG is triple S vð Þ¼ <sup>0</sup> , a0, R, v<sup>0</sup> ., where a<sup>0</sup> and R are, as higher, the title object and scheme, respectively, and v<sup>0</sup> is the multiset, which may contain nonterminal multiobjects, used for elimination of the number of generation steps. So, this version of UMG has specific semantics, which fully corresponds to the sense of order completion by the use of aforementioned RB.

The main difference of UMG RG from UMG is that they generate not multisets, but pairs , v, v<sup>0</sup> . , where v is the MS, created while previous generations steps, and v<sup>0</sup> is the rest of RB, which may be used at the next such step.

If there is a non-terminal multiobject n � a in multiset v, and at the same times MS v<sup>0</sup> includes MO n<sup>0</sup> � a, then following action depends on the relation between n and n<sup>0</sup> . If n . n<sup>0</sup> , then there are already n<sup>0</sup> OR a in the resource base, and there is no any need to manufacture them—it is sufficient to manufacture n � n<sup>0</sup> OR a and eliminate n<sup>0</sup> OR a from v<sup>0</sup> to represent that they are already used while order completion. If n<sup>0</sup> ≤n, then all necessary OR a are already in the RB, and there is no need in generation here at all; it is sufficient to subtract f g n � a , so there would be MO n � n<sup>0</sup> ð Þ� a in the RB after this action, because n OR a are spent (if n<sup>0</sup> ¼ n, there would be no OR a in the RB).

Formal definition of semantics of UMG RG S vð Þ¼ <sup>0</sup> , a0, R, v<sup>0</sup> . , i.e., a set of relations, describing generation of a set VS vð Þ <sup>0</sup> of pairs , v, v<sup>0</sup> . , is as follows:

$$\overline{V}\_{(0)} = \{ \,\, \lhd \{ \mathbf{1} \cdot \mathbf{a}\_0 \}, \mathbf{v}\_0 \rhd \}, \tag{19}$$

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

$$V\_{(i+1)} = V\_{(i)} \cup \left( \bigcup\_{<\eta, \, \nu' \ge \, : \, \Gamma \nmid \, \nu \in R} \bigcup\_{r \in R} \{\rho(\nu, \nu', r)\} \right), \tag{20}$$

φ v; v<sup>0</sup> ð ; , a ! n<sup>1</sup> � a1; …; nm � am . Þ ¼ ¼ , v � f g n � a , v<sup>0</sup> � f g n � a . , if n � a ∈ v & n<sup>0</sup> � a ∈ v<sup>0</sup> & n<sup>0</sup> ≥n , v � f g n � a þ n � n<sup>0</sup> ð Þ ∗ f g n<sup>1</sup> � a1; …; nm � am , v<sup>0</sup> � n<sup>0</sup> f g � a . , if n � a ∈ v & n<sup>0</sup> � a ∈ v<sup>0</sup> & n<sup>0</sup> 6¼ 0 & n<sup>0</sup> , n∨ n � a ∈ v & n<sup>0</sup> ¼ 0 8 >>>>>>>>< >>>>>>>>: (21)

f g ∅ otherwise,

Concerning affected active resources, it is reasonable to underline that NHI may destroy them up to unrecoverable state (this may be represented by inclu-

unoperational, but reparable, state, that may be represented by inclusion to Δv

By this we finish a short survey of known results on resilience of industrial systems. Before we move to sociotechnological systems, let us generalize the intro-

As seen, both introduced criteria of IS resilience operate only terminal resources, which are used by IS for production of amounts of OR, being the goal of order. By this, they trivially repeat criterion of order completeness (12) with the only replacement of the IS initial resource base by RB, reduced by

However, if to take into account that there may be some non-terminal OR, already manufactured by IS during time interval between the start of order completion and moment of NHI, it would be sensible to consider these OR during recognition of IS resilience, or, in the other words, to generalize notion of resource base, including to RB not only terminal, but also non-

But, evidently, this generalization makes the introduced criteria non-applicable. Let us propose correct criterion for the case of RB, containing not only terminal but

For this purpose we propose here so-called unitary multiset grammars with

UMG RG is triple S vð Þ¼ <sup>0</sup> , a0, R, v<sup>0</sup> ., where a<sup>0</sup> and R are, as higher, the title object and scheme, respectively, and v<sup>0</sup> is the multiset, which may contain nonterminal multiobjects, used for elimination of the number of generation steps. So, this version of UMG has specific semantics, which fully corresponds to the sense of

The main difference of UMG RG from UMG is that they generate not multisets, but pairs , v, v<sup>0</sup> . , where v is the MS, created while previous generations steps, and

, then there are already n<sup>0</sup> OR a in the resource

Vð Þ <sup>0</sup> ¼ f g , f g 1 � a<sup>0</sup> ; v<sup>0</sup> . , (19)

If there is a non-terminal multiobject n � a in multiset v, and at the same times MS v<sup>0</sup> includes MO n<sup>0</sup> � a, then following action depends on the relation

manufacture n � n<sup>0</sup> OR a and eliminate n<sup>0</sup> OR a from v<sup>0</sup> to represent that they are already used while order completion. If n<sup>0</sup> ≤n, then all necessary OR a are

Formal definition of semantics of UMG RG S vð Þ¼ <sup>0</sup> , a0, R, v<sup>0</sup> . , i.e., a set of relations, describing generation of a set VS vð Þ <sup>0</sup> of pairs , v, v<sup>0</sup> . , is as

base, and there is no any need to manufacture them—it is sufficient to

already in the RB, and there is no need in generation here at all; it is sufficient to subtract f g n � a , so there would be MO n � n<sup>0</sup> ð Þ� a in the RB after this action, because n OR a are spent (if n<sup>0</sup> ¼ n, there would be no OR a

sion to <sup>Δ</sup><sup>v</sup> MO <sup>N</sup> � <sup>t</sup>‐r=z) or, in the better case, transfer them to the

MO <sup>n</sup><sup>0</sup> � <sup>t</sup>‐r=z, where <sup>n</sup><sup>0</sup> is less than multiplicity <sup>n</sup> of OR <sup>t</sup>‐r=<sup>z</sup> <sup>∈</sup> <sup>v</sup>.

Natural Hazards - Risk, Exposure, Response, and Resilience

3. Generalized criterion of resilience of industrial systems

duced criteria.

NHI.

terminal OR.

also non-terminal OR.

between n and n<sup>0</sup>

in the RB).

follows:

244

reduced generation (UMG RG).

order completion by the use of aforementioned RB.

. If n . n<sup>0</sup>

v<sup>0</sup> is the rest of RB, which may be used at the next such step.

$$
\overline{V}\_{S(v\_0)} = V\_{(\infty)}.\tag{22}
$$

This definition fully corresponds to the previous verbal description and is similar to (5)–(8). The mission of function φ, defined by (21), is the same as the mission of function π, defined by (7). Some comments would be done to its second alternative, namely, the case where multiset v<sup>0</sup> does not contain OR a at all (or, just the same, multiplicity n<sup>0</sup> of multiobject n<sup>0</sup> � a, entering v<sup>0</sup> , is zero); this is equivalent to a more general case, when n<sup>0</sup> � a ∈ v<sup>0</sup> and non-zero multiplicity n<sup>0</sup> is less than n. As seen, the result of subtraction of the empty multiset n<sup>0</sup> f g � a , where n<sup>0</sup> ¼ 0, from multiset v<sup>0</sup> , is unchanged v<sup>0</sup> , and this branch of (21) is just the same, as the first alternative of (7).

The introduced UMG RG provide formulation of the generalized criterion of IS resilience.

Let q ¼ n<sup>1</sup> � b1; …; nl f g � bl be order, R—technological base of the industrial system, and v—its resource base, such that

$$
\beta(v) \subseteq A\_s. \tag{23}
$$

(That is, it contains not only terminal but also non-terminal OR). Consider UMG RG Sqð Þ¼ v , tbq, Rq, v . , where

Rq ¼ R ∪ , tbq ! n<sup>1</sup> � b1; …; nl f g � bl . : (24)

(Here, UR is written in the angle brackets for unambiguity.)

Statement 4. Order q to IS with technological base R and resource base v may be completed, if

$$\left(\exists \ \leq \overline{v}, \overline{v}\prime > \in \overline{V}\_{\mathcal{S}\_{\mathbb{Q}}(v)}\right) \overline{v} \subseteq \overline{v}\prime.\mathbb{L}\tag{25}$$

As seen, if RB does not contain non-terminal OR, (25) and (13) are equivalent. As higher, RB, relevant to criterion 4, is called sufficient for order q completion by IS. Evidently, v<sup>0</sup> � v is RB, remained after completion of order q.

Example 4. Let S<sup>0</sup> ¼ , aircraft, R<sup>0</sup> . be as in Example 1, order q ¼ f g 4 � aircraft , and resource base of the industrial system is v ¼ f6 � fuselage; 12 � wing; 300 � kW; 800 � mnt � asm � aircraft; 1100000 � usdg:

As seen,

$$S\_q(v) = \le tbq, R\_q, v \ge 1$$

where

$$R\_q = R' \cup \{  \},$$

and resource base v contains non-terminal multiobject 12 � wing, which means 12 wings are already manufactured and ready to be mounted to fuselages in order to make aircrafts.

According to (19)–(21), VSq ð Þ¼ v , v; v f g <sup>0</sup> . , where

v ¼ f g 4 � fuselage; 40 � kW; 640 � mnt � asm � aircraft; 600000 � usd ,

v<sup>0</sup> ¼ f g 6 � fuselage; 4 � wing; 300 � kW; 800 � mnt � asm � aircraft; 1100000 � usd :

Because v⊂ v<sup>0</sup> , order q may be completed by IS due to the number of already manufactured wings, which is greater than the required for manufacturing of four aircrafts. ∎

Let resource base v be sufficient for order q completion by IS with technological base R, and Δv is NHI on this system.

Statement 5. IS, completing order q, is resilient to NHI Δv, if

$$\left(\exists \ \leq \overline{v}, \overline{v}\prime > \in \overline{V}\_{\mathbb{S}\_{\mathbb{Q}}(v-\Delta v)}\right) \overline{v} \subseteq \overline{v}\prime. \tag{26}$$

v ¼ f3 � fuselage=z1, 2 � wing=z1, 4 � frame=z2, 5 � engine=z2, 8 � wheel=z2, 500 � l � petrol=z2, 1 � wing=z2, 100 � vel=z2, 10000 � l � petrol=z3,

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

as four frames, five engines, and eight wheels, which may be used for

back without repair.

VSqð Þ<sup>v</sup> ¼ f g , f g 2 � aircraft=z<sup>1</sup> ; v .

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

v � f g 3 � wing=z<sup>1</sup> . g

v � f g 3 � wing=z<sup>1</sup> . g

latter is sufficient for this order completion.

(19)–(21):

Because

247

manufacturing of some additional number of wings. Moreover, there is transportation vehicle vel at z2, which may remove ready wings from z<sup>2</sup> to z1, and also 500 liters of petrol for refueling this vehicle. But, as seen, this amount of petrol is not sufficient for the relocation of two wings from z<sup>2</sup> to z1. So, the required amount of petrol, i.e., 500 liters, must be removed by the pipeline to z<sup>2</sup> from z3, where petrol storage is located, containing at the current moment 10,000 liters of this fuel. Multiobjects 100000 � link=z<sup>3</sup> and 100000 � pump=z<sup>3</sup> represent the technical state of petrol pipeline link and pump, which is sufficient for execution of 100,000 working cycles, each providing removal of 1 liter of petrol from z<sup>3</sup> to z2. Similarly, MO 100 � vel=z<sup>2</sup> represents the technical state of the vehicle, which is able to make 100 transportation cycles from z<sup>2</sup> to z<sup>1</sup> and

This verbal description makes it evident, how in fact order completion may be planned by UMG RG application. Let us consider how it is really done according to

¼ f g , f g 2 � fuselage=z1; 1 � wing=z1; 20 � kW=z<sup>1</sup> ; v � f g 3 � wing=z<sup>1</sup> .

¼ f , f g 2 � fuselage=z1; 1 � wing=z2; 1000 � l � petrol=z2; 1 � vel=z<sup>2</sup> ,

¼ f , f2 � fuselage=z1, 20 � kW=z1, 1 � frame=z2, 1 � engine=z2,

¼ f , f2 � fuselage=z1, 20 � kW=z1, 1 � frame=z2, 1 � engine=z2,

f2 � fuselage=z1, 20 � kW=z1, 1 � frame=z2, 1 � engine=z2, 4 � wheel=z2, 12 � kW=z2,

9500 � l � petrol=z3, 100000 � link=z3, 100000 � pump=z3, 200 � kW=z3g,

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

order q is completed by DIS with technological base R and resource base v; the

500 � l � petrol=z3, 500 � link=z3, 500 � pump=z3, 1 � kW=z3g

4 � wheel=z2, 12 � kW=z2, 500 � l � petrol=z3, 500 � link=z3,

500 � pump=z3, 1 � kW=z3g, v � f g 3 � wing=z1; 500 � l=petrol=z<sup>2</sup> . g:

4 � wheel=z2, 12 � kW=z2, 1000 � l � petrol=z2, 1 � vel=z2g,

¼ f g , f g 2 � fuselage=z1; 4 � wing=z1; 20 � kW=z<sup>1</sup> ; v .

100000 � link=z3, 100000 � pump=z3, 50 � kW=z1, 150 � kW=z2, 200 � kW=z3g: This means that at location z<sup>1</sup> there are three fuselages, ready to be mounted with wings, but there are only two wings at this location, so two more wings, necessary for the production of two aircrafts, must be removed to z<sup>1</sup> from z2. However, there are two ready wings at z2; there is only one such wing, as well

Otherwise, this IS is vulnerable to this NHI. ∎

This criterion may be generalized on distributed IS in the same manner, as it was done in [2] and described in the previous section.

Let RB v be sufficient for order q completion by DIS with TB R, and Z is NHI on this system.

Statement 6. DIS, completing order q, is resilient to NHI Z, if

$$\left(\exists \, \leq \overline{v}, \overline{v}\, \middle| \, \leq \, \in \overline{V}\_{\mathbb{S}\_{q}(v-\Delta v(Z))}\right) \overline{v} \leq \overline{v}\,. \tag{27}$$

Otherwise, this DIS is vulnerable to this NHI. ∎

It is clear that DIS RB contains both terminal and non-terminal objects, located at various places.

Example 5. Let DIS be represented by UMG S ¼ , aircraft=z1, R . , where R contains the following unitary rules:

$$\begin{aligned} \textit{aircr} & \textit{get/z}\_1 \rightarrow \textbf{1} \cdot \textit{fuselag/z}\_1, \textbf{2} \cdot \textit{wing/z}\_1, \textbf{10} \cdot \textit{kW/z}\_1, \\\\ \textit{bwing/z}\_2 & \rightarrow \textbf{1} \cdot \textit{fuse/z}\_2, \textbf{1} \cdot \textit{engine/z}\_2, \textbf{4} \cdot \textit{wheel/z}\_2, \textbf{12} \cdot \textit{kW/z}\_2, \\\\ \textit{bwing/z}\_1 & \rightarrow \textbf{1} \cdot \textit{wing/z}\_2, \textbf{10} 0 0 \cdot \textit{l} - \textit{petal/z}\_2, \textbf{1} \cdot \textit{vel/z}\_2, \\\\ \textit{l} & \rightarrow \textbf{1} \cdot \textit{petal/z}\_2 \rightarrow \textbf{1} \cdot \textit{l} - \textit{petal/z}\_3, \textbf{1} \cdot \textit{l} \,\textbf{kW/z}\_3, \textbf{1} \cdot \textit{pumped/z}\_3, \textbf{0} \,\textbf{0} \,\textbf{1} \cdot \textit{kW/z}\_3. \end{aligned}$$

Here, the first two UR are slightly modified versions of technological base, described by UMG S; the only difference is that all OR are composites, including names of locations. As seen, aircrafts are assembled at place z1, while wings—at place z2. The third UR defines that to remove one wing to z<sup>1</sup> from z2, some transportation vehicle vel must be used, and also 1000 liters of petrol for its refueling, necessary for wing removal to z<sup>1</sup> and return to z2. At last, the fourth UR defines that to transport petrol from place z3, where it is stored, there is used pipeline fragment, consisting of link and pump, the latter consuming 0.001 kW of electrical energy to remove 1 liter of petrol from z<sup>3</sup> to z2. Assembling one aircraft and one wing is also an energy-consuming operation that is represented by multiobjects 10 � kW=z<sup>1</sup> and 12 � kW=z2, having placed in the bodies of the first and the second UR, respectively.

Let order q ¼ f g 2 � aircraft=z<sup>1</sup> , and resource base of DIS is

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

v ¼ f3 � fuselage=z1, 2 � wing=z1, 4 � frame=z2, 5 � engine=z2, 8 � wheel=z2, 500 � l � petrol=z2, 1 � wing=z2, 100 � vel=z2, 10000 � l � petrol=z3, 100000 � link=z3, 100000 � pump=z3, 50 � kW=z1, 150 � kW=z2, 200 � kW=z3g:

This means that at location z<sup>1</sup> there are three fuselages, ready to be mounted with wings, but there are only two wings at this location, so two more wings, necessary for the production of two aircrafts, must be removed to z<sup>1</sup> from z2. However, there are two ready wings at z2; there is only one such wing, as well as four frames, five engines, and eight wheels, which may be used for manufacturing of some additional number of wings. Moreover, there is transportation vehicle vel at z2, which may remove ready wings from z<sup>2</sup> to z1, and also 500 liters of petrol for refueling this vehicle. But, as seen, this amount of petrol is not sufficient for the relocation of two wings from z<sup>2</sup> to z1. So, the required amount of petrol, i.e., 500 liters, must be removed by the pipeline to z<sup>2</sup> from z3, where petrol storage is located, containing at the current moment 10,000 liters of this fuel. Multiobjects 100000 � link=z<sup>3</sup> and 100000 � pump=z<sup>3</sup> represent the technical state of petrol pipeline link and pump, which is sufficient for execution of 100,000 working cycles, each providing removal of 1 liter of petrol from z<sup>3</sup> to z2. Similarly, MO 100 � vel=z<sup>2</sup> represents the technical state of the vehicle, which is able to make 100 transportation cycles from z<sup>2</sup> to z<sup>1</sup> and back without repair.

This verbal description makes it evident, how in fact order completion may be planned by UMG RG application. Let us consider how it is really done according to (19)–(21):

$$\begin{split} \overline{V}\_{\overline{\nabla}\_{\overline{\nabla}\_{\overline{\nabla}}} &= \{ < \{ 2 \cdot \text{fracr} \text{deg}/\text{z}\_1, \cdot v > \} \} \\ &= \{ < \{ 2 \cdot \text{fraclage} /\text{z}\_1, \cdot \text{wing}/\text{z}\_1, 20 \cdot kW/\text{z}\_1 \}, v > \} \\ &= \{ < \{ 2 \cdot \text{fraclage} /\text{z}\_1, 1 \cdot \text{wing}/\text{z}\_1, 20 \cdot kW/\text{z}\_1 \}, v - \{ 3 \cdot \text{wing}/\text{z}\_1 \} > \} \\ &= \{ < \{ 2 \cdot \text{fraclage} /\text{z}\_1, 1 \cdot \text{wing}/\text{z}\_1, 1000 \cdot l - \text{prod}/\text{z}\_1, 1 \cdot \text{vol}/\text{z}\_2 \}, \\ &v - \{ 3 \cdot \text{wing}/\text{z}\_1 \} > \\ &= \{ < \{ 2 \cdot \text{fraclage} /\text{z}\_1, 10 \cdot kW/\text{z}\_1, 1 \cdot \text{frac} /\text{mage}/\text{z}\_2, 1 \cdot \text{engine}/\text{z}\_2 \}, \\ &4 \cdot \text{wole}/\text{z}\_2, 12 \cdot kW/\text{z}\_2, 1000 \cdot l - \text{prod}/\text{z}\_2, 1 \cdot \text{vol}/\text{z}\_2 \}, \\ &v - \{ 3 \cdot \text{wing}/\text{z}\_1 \} > \\ &= \{ < \{ 2 \cdot \text{frac{lage} /$$

Because

and resource base v contains non-terminal multiobject 12 � wing, which means 12 wings are already manufactured and ready to be mounted to fuselages in order to

v<sup>0</sup> ¼ f g 6 � fuselage; 4 � wing; 300 � kW; 800 � mnt � asm � aircraft; 1100000 � usd :

manufactured wings, which is greater than the required for manufacturing of four

Let resource base v be sufficient for order q completion by IS with technological

. ∈ VSqð Þ <sup>v</sup>�Δ<sup>v</sup>

This criterion may be generalized on distributed IS in the same manner, as it was

Let RB v be sufficient for order q completion by DIS with TB R, and Z is NHI on

. ∈ VSqð Þ <sup>v</sup>�Δv Zð Þ

It is clear that DIS RB contains both terminal and non-terminal objects, located

Example 5. Let DIS be represented by UMG S ¼ , aircraft=z1, R . , where R

l � petrol=z<sup>2</sup> ! 1 � l � petrol=z3, 1 � link=z3, 1 � pump=z3, 0:001 � kW=z3: Here, the first two UR are slightly modified versions of technological base, described by UMG S; the only difference is that all OR are composites, including names of locations. As seen, aircrafts are assembled at place z1, while wings—at place z2. The third UR defines that to remove one wing to z<sup>1</sup> from z2, some transportation vehicle vel must be used, and also 1000 liters of petrol for its refueling, necessary for wing removal to z<sup>1</sup> and return to z2. At last, the fourth UR defines that to transport petrol from place z3, where it is stored, there is used pipeline fragment, consisting of link and pump, the latter consuming 0.001 kW of electrical energy to remove 1 liter of petrol from z<sup>3</sup> to z2. Assembling one aircraft and one wing is also an energy-consuming operation that is represented by multiobjects 10 � kW=z<sup>1</sup> and 12 � kW=z2, having placed in the bodies of the first and the second

, order q may be completed by IS due to the number of already

v⊆v<sup>0</sup>

v⊆v<sup>0</sup>

: (26)

: (27)

According to (19)–(21), VSq ð Þ¼ v , v; v f g <sup>0</sup> . , where

Natural Hazards - Risk, Exposure, Response, and Resilience

v ¼ f g 4 � fuselage; 40 � kW; 640 � mnt � asm � aircraft; 600000 � usd ,

Statement 5. IS, completing order q, is resilient to NHI Δv, if

Statement 6. DIS, completing order q, is resilient to NHI Z, if

∃ , v; v<sup>0</sup>

aircraft=z<sup>1</sup> ! 1 � fuselage=z1, 2 � wing=z1, 10 � kW=z1,

wing=z<sup>1</sup> ! 1 � wing=z2, 1000 � l � petrol=z2, 1 � vel=z2,

Let order q ¼ f g 2 � aircraft=z<sup>1</sup> , and resource base of DIS is

wing=z<sup>2</sup> ! 1 � frame=z2, 1 � engine=z2, 4 � wheel=z2, 12 � kW=z2,

Otherwise, this DIS is vulnerable to this NHI. ∎

∃ , v; v<sup>0</sup>

Otherwise, this IS is vulnerable to this NHI. ∎

done in [2] and described in the previous section.

make aircrafts.

Because v⊂ v<sup>0</sup>

base R, and Δv is NHI on this system.

aircrafts. ∎

this system.

at various places.

UR, respectively.

246

contains the following unitary rules:

$$\begin{aligned} & \{2 \cdot fusleq/z\_3, 20 \cdot kW/z\_3, 1 \cdot frame/z\_3, 1 \cdot engine/z\_3, 4 \cdot wheel/z\_3, 12 \cdot kW/z\_2, \\ & 500 \cdot l - petrol/z\_3, 500 \cdot link/z\_3, 500 \cdot pump/z\_3, 1 \cdot kW/z\_3\} \\ & \subset \{3 \cdot fusleq/z\_3, 50 \cdot kW/z\_3, 4 \cdot frame/z\_3, 5 \cdot engine/z\_3, 8 \cdot wheel/z\_2, 150 \cdot kW/z\_2, 1 \cdot Hz/z\_3, 1 \cdot Hz/z\_3, 1 \cdot Hz/z\_3, 1 \cdot Hz/z\_3, 1 \cdot Hz/z\_3, 1 \cdot Hz/z\_3\}, \\ & 5950 \cdot l - petrol/z\_3, 100000 \cdot link/z\_3, 100000 \cdot pump/z\_3, 200 \cdot kW/z\_3\}, \end{aligned}$$

order q is completed by DIS with technological base R and resource base v; the latter is sufficient for this order completion.

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

v � Δv Zð Þ

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

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 individ-

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

that means there are m<sup>1</sup> structures (of type) str1, …, mk structures (of type) strk;

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 ! n<sup>1</sup> � pstn1, np � pstnp, m<sup>1</sup> � str1, …, ms � strs,

l<sup>1</sup> � tech1, …, lt � techt,

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

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 fol-

the latter case corresponding to the fully robotized (unmanned) system. Every techi, in turn, may be decomposed recursively until terminal objects, which names

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

person ! n<sup>1</sup> � person1, …, nl � personl

ri � res<sup>i</sup> ri , m<sup>i</sup> <sup>1</sup> � devi

have been placed only in the bodies of unitary rules.

personi ! ki

consumed resources. This may be represented by unitary rule

<sup>1</sup> � res<sup>i</sup>

<sup>1</sup>, …, k<sup>i</sup>

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

str ! n<sup>1</sup> � pstn1, …, np � pstnp, l<sup>1</sup> � tech1, …, lt � techt (31)

tech ! n<sup>1</sup> � pstn1, …, np � pstnp, l<sup>1</sup> � tech1, …, ls � techs, (33)

tech ! l<sup>1</sup> � tech1, …, ls � techs, (34)

<sup>1</sup>, …, m<sup>i</sup>

li � devi li

, (35)

, (36)

str ! n<sup>1</sup> � pstn1, …, np � pstnp, (32)

structures ! m<sup>1</sup> � str1, …, mk � strk, (29)

(30)

uals, not entering any of the aforementioned structures. Object structures is the head of the single unitary rule

if any str i of str1, …, strk is unique, then mi ¼ 1.

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

multigrammatical representation is like

or

lows:

and

249

providing their operation.

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

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

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

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 system with resource base, reduced by this NHI.

Before we move to further discourse, let us clarify interconnections between basic notions, which will be used below.

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

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

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