Abstract

The chapter describes multiset-based approach to the assessment of resilience/ vulnerability of the distributed sociotechnological systems (DSTS) to natural hazards (NH). DSTS contain highly interconnected and intersected consuming and producing segments, and also resource base (RB), providing their existence and operation. NH impacts may destroy some local elements of these segments, as well as some parts of RB, thus initiating multiple chain effects, leading to negative consequences far away from the NH local strikes. To assess DSTS resilience to such impacts, multigrammatical representation of DSTS is used. A criterion of DSTS sustainability to NH, being generalization of similar criterion, known for industrial (producing) systems, is proposed. Application of this criterion to critical infrastructures is considered, as well as solution of the reverse problem, concerning subsystems of DSTS, which may stay functional after NH impact.

Keywords: resilience and vulnerability, natural hazards, sociotechnological systems, critical infrastructures, multisets, multiset grammars, unitary multiset grammars

### 1. Introduction

Modern large-scale distributed sociotechnological systems (DSTS) include anthropogenic and technogenic components, i.e., humans and various technical devices, respectively, operating in common in order to provide sufficient quality of life to humans, and this sufficiency may be defined by some threshold amounts of resources, consumed by them during some fixed period of life. These resources, in turn, must be produced and relocated from places of their production to places of their consumption by application of the aforementioned devices and their aggregates. The last also uses specific resources, necessary for their operation.

By this, every DSTS may be represented as composition of two segments consuming and producing (both containing humans and devices)—and resource base, which provides their existence and operation. These segments are highly interconnected and intersect, because a large number of humans and devices are consumers and creators of resources simultaneously.

Natural hazard impacts (NHI) may destroy some local elements of the aforementioned segments and resource base, and this destruction initiates multiple chain (or cascading) effects, caused by the absence or lack of resources,

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

2. Assessment of resilience of industrial systems

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

for i 6¼ j ai 6¼ aj. Set

grammar") [12, 13].

of UR

n0 <sup>1</sup> � a<sup>0</sup>

rules:

239

<sup>1</sup>, …, n<sup>0</sup>

<sup>k</sup> � a<sup>0</sup> k.

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

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

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

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

UMG is a couple S ¼ , a0, R . , where a<sup>0</sup> is called title object and R is called

where object a is called head and list n<sup>1</sup> � a1, …, nm � am—body of this UR. List is

The so-called structural and technological interpretations of unitary rules are

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

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

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

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

aircraft ! 1 � fuselage, 2 � wing, wing ! 1 � frame, 1 � engine, 4 � wheel: 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>

<sup>1</sup> � a<sup>0</sup>

<sup>1</sup>, …, n<sup>0</sup>

<sup>k</sup> � a<sup>0</sup>

<sup>k</sup> (4)

. ,

a ! n<sup>1</sup> � a1, …, nm � am, n<sup>0</sup>

scheme, being the set of unitary rules (UR), having the form

interpreted as multiset, i.e., f g n<sup>1</sup> � a1; …; nm � am .

used in the IS resilience assessment [2].

"object/resource," abbreviated OR).

where R<sup>0</sup> contains the following two URs:

v ¼ f g n<sup>1</sup> � a1; …; nm � am , (1)

βð Þ¼ v f g a1; …; am (2)

a ! n<sup>1</sup> � a1, …, nm � am, (3)

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 DSTS).

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

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 impact.

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 there.

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 the conclusion.

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