AACIMP 2010 Summer School lecture by Alexander Makarenko. "Applied Mathematics" stream. "General Tasks and Problems of Modelling of Social Systems. Problems and Models in Sustainable Development" course. Part 9. More info at http://summerschool.ssa.org.ua

1. Sustainable Development
Formal Definition and
Modeling
Alexander Makarenko
Institute for applied systems analysis
NTUU „KPI”, Prospect Pobedy 37, 03056,
Kiev-56, Ukraine
makalex@i.com.ua

2. ABSTRACT
 The general questions of sustainable development - the conception,
examples and limitations of implementation are considered.
 General principles for formalization of notion of sustainable
development are discussed.
 Different models are considered for study of sustainable
development.
 Relation stabilitytransformation had been investigated.
 The implementation of transformation process in large socio –
economic systems as the application of operational researches.

 The consideration of mental aspects of SD.

3. I. COMMON DEFINITIONS OF
SUSTAINABLE DEVELOPMENT
 The concept of sustainable development has a long history of its essential
components. First of all it needs to remark many natural science investigations and
ecology. Another part is demography (may be since the works of Maltus). First
explicit implementation and first working tool for considering SD is system dynamics
since the work of Forester, Meadows and others (Iscvt, 2002; MIT, 2002). Important
role also has played the concepts and the models of the World by I. Wallerstain, B.
Fuller, A.Frank. One of the past focal point of SD was world leaders conference at
Rio- de Janeiro (1992), and last Summit at Johannesbourg (2002), where some
definitions and prospects had been formulated.
 But the experience since 1992 had followed to necessity of further improvement in
concepts (see many conferences: (Euroscience, 2002) and others). There are many
reasons for this. Of course main is permanent changes in recent world closely
connected to global processes. But it exists some intrinsic problems in SD which
force to further development of approach. First of all the main definition of sustainable
development is verbal and conventional. A little number of quantitative approach exist
(system dynamics and some types models for modeling large- scales processes -
(Spangenberg et al., 2001) and multi- agent approach (Beckenbach, (2001); Iscvt,
(2002)), indexes (Zgurovski&Gvishiani, 2008). The leak of full operational models
follows the shortage of sustainable development indexes (Spangenberg et al., 2001;
Iscvt, (2002)) which needs for practical planning.

4. Basic description of SD
 “"Sustainabledevelopment is development that meets the needs of the
present without compromising the ability of future generations to meet their
own needs.
 It contains within it two key concepts:
 • the concept of 'needs', in particular the essential needs of the world's poor,
to which overriding priority should be given; and
 • the idea of limitations imposed by the state of technology and social
organization on the environment's ability to meet present and future needs."
 - following G.H.Bruntland Commission (1987)
 That is there exists discrepancy between the natural resources and between determined by economics and history
way of their exploitation.

5. Some factors in SD process

6. Usual representations of
parameters in SD processes

7. Trajectories of system in case of
fatal restrictions of resources

8. Possibilities of sustainable
trajectories

9. Case of possible change of leading
resources

10. II. FORMALIZAQTION OF
SUSTAINABILITY AND SD
 Necessary components for SD:
 Resources
 Restrictions on resources
-- Evolutionary aspects
 Goal of the system
 Existing of many generations
 Indexes of sustainability
-- Decision – making
 Mental properties of peoples and cultural aspects
 Environment
 Technology …

11. Parameters and structures in SD(1)
 Therefore in a next section we will make an effort give some primary
considering to formalization of the SD notion.
 At first we will indicate structures and notions which it follows to take into
account in a task about local SD.
 1. We will designate the parameters of the system and their description
(external, internal, managers and etc.) Set of parameters {Par}. We in this
subsection will not consider in the details of property of these (and elements
of description the following), and only we will make an effort select, what
structures it follows to consider. So, for example, we without the special
necessity will not consider possible metrical and topology structures,
ordering relations, symmetries and dr. on the great number {Par}.
 2. Equations which describe the systems and processes {Equat}.
 3. Set of trajectories of the systems {Traj}.
 4. Limitations on trajectories and parameters of the system – set {Ώ} and
 set of boundaries of limitations {∂Ώ}.
 5. Set of criteria of sustainability {SCrit} or the SD criteria {SDCrit}.

12. Parameters and structures in SD(2)
 6. Set external handling parameters {Contr}.
 7. Set presenting the age structure of population on an interval of time [0, T]
{Age[0, T]}. If an interval is not indicated obviously, we will write {Age}.
 8. Set of initial conditions {Init}
 9. Structure of the system {StSys}, structure of processes {StProc} and
structure of individuals {StInd}. These great numbers can be entered, if
indeed it is known as such objects are arranged. But it is possible
confidently to assume that such structures indeed exist (even if obviously
about them nothing is known).
 10. Additional requirements to the components (desirable) – additional to
obligatory limitations {Ώ} and {∂Ώ}. We will designate them {Aux}.
 11. Descriptions of process of acceptance of decisions. We will designate
those {Decis}. In the case of necessity it is possible to dash them on
separate components.
 12. Set of nondefinetness (uncertainties) in the system {NonDef}. In the
case of necessity this set also can be broken-down on components.

13. Description of task
 We will describe now, what does it mean SD.
 Definition. The SD task. To find such objects from
 {Init}, {St = {StSys} ® {StProc} ® {StInd}}, {Contr}, {Decis},
 such, that as a result such trajectory at evolution of the
system turns out
 tr € {Traj}, that is executed <tr, Cr>t € {SDCrit} ® {Aux}

 where<tr, Cr>t means the calculation of value of the SD
criterion in a moment
 t on a trajectory tr, thus the results of calculation must at
every instant to lie in {SDCrit} and in {Aux}.

14. Parameters and structures in SD(3)
 We will show sense of determination for the best idea on simple illustrations
as geometrical pictures.
 We will do a few remarks here.
 The remark 1. To 12 indicated higher structures it is possible to add another
one – set of models {Models}, if we use the modeling.
 The remark 2. At consideration of different systems it is possible in such
raising to take category approach and try to select the category of the
systems with the steady development CatSD.
 The remark 3. Taking into account the possible multivaluedness of
trajectories of the system (which can arise up from different reasons,
including because SNET – includes a social component). Therefore in place
of one trajectory tr it is possible to enter formulations with the bunch of
trajectories Ptr.
 The remark 4. It is possible also existence of fluctuations and other
uncertainties (by the way, this is essential at estimation of risks). Then it is
possible to take into account vagueness in objects, considering some sets
from 1-13 to
 {SDCrit}(NonDef), {Aux}(Nondef), {St}(NonDef).

15. Parameters and structures in SD(4)
 Meaning the structure of Definition 1 for the SD task it is possible already to move
farther and extend and at the same time go into detail determination.
 Consideration of presence of many different generations (for simplicity below we
speak about two generations), for example, maybe, that at two generations different
criteria SD, then
 {SDCrit} = {SDCrit}(Generation 1) ® {SDCritGeneration 2}
 There can be different for different generation’s limitations on the managements
 {RCtrl} = {RCtrl}(Generation 1) ® {RCtrl}(Generation 2)
 It can be in principle, that
 {SDCrit} = {SDCrit} (Generation 1) ® {MSDCrit} (Generation 2)
 where {MSDCrit}(Generation 2) is the set of possible criteria of SD for second
generation.
 We cannot know exactly {MSDCrit}(Generation 2) by virtue of that it relies on future
technologies, and actually from future knowledge {Knowl}(Generation 2)
 about which we can only build guess-work (we consider that great number of current
knowledge of the given generation {Knowl}(Generation 1) in the first approaching it is
known – or, for example, this great number of realized in technologies knowledge).

16. Examples (0)
 We will indicate some examples for illustration.
 The example 1. We can in the first approaching set the establish set {SDCrit} as the
set, where some model indexes are satisfied to the accepted pictures of SD. That is,
if indexes ind, which belong set {Ind} of indexes answering the pictures of experts of
SD – to the set {IndSD}.
 Then a process will be with SD, if ind € {Ind}, thus ind € {IndSD} V t.
 Thus, of course, many thin details of processes and conducts of the systems are not
taken into account. We will mark in passing some interesting works devoted to the
search of the SD indexes.
 By the way, in works D.Forrestera knowledge does not enter in the complement of
basic variables obviously.
 The example 2. (The Chichilniska works). In them, as far as it is possible to judge, is
formed on the economic considering only.
 The example 3. (The Rand work). {St} changes in them in course of time, and {SDCrit}
it is possible to formulate mathematically strictly.
 The example 4. The cases are not eliminated, when {SDCrit} can be formed by the
Lyapunov’s function.
 The example 5. Evolutional economy. It is possible, in principle, to make table of
comparison of the SD researches in such formalization

17. Examples (2)
 The offered chart of consideration befits, looks like, to any variant of the SD
processes (both descriptive (verbal), and concrete practical tasks or for tasks already
having formalization as mathematical tasks.
 Review and comparison of the mathematical raising was very useful, and in the future
we hope also to present their results as table. Here we will make illustrative examples.
 The Example 1. (Local SD). More simple case, if a situation does not change
substantially, simple models (as a rule, this proper one of «pillars») of the system and
without consideration of change of generations
 It is clear, that it is possible to write down the SD criteria different (for example, speed
on the trajectories, stocked energies for the management) and others.
 In such kind it reminds the tasks of traffic control on the mathematical raising (traffic
control with limitation).
 By the way, the guided Markov’s chains from the given point of view, and then the
Markov’s chains with anticipation or with a vagueness – semi-Marko’s.
 It is possible the same for Lorent’s (for chaotic trajectories) – the SD loud speakers
of chaos.
 It is possible similarly to the same stochastic differential equalizations in partial
derivative – here at the level of communication with synergetic and dissipative
structures.

18. Simple examples (1)

19. Simple examples (2)

20. III. SD MODELING AND
INTERPRETATIONS

21. 1D ‘presentation’ for two attractors
case

22. Attracting with restrictions
accounting

23. Transitions between attractors with
restrictions

24. IV. MENTALITY AND EDUCATION
 All such issues are useful for considering sustainable development. The
problem of sustainable development in our approach looks like the
problems of the evolution of the system in the terms of attractors, and the
transitions from one attractor to other (Makarenko, 2003).
 For understanding and management of sustainable development the use of
the concept of 'landscape' presented in the Section 2 (see for example
Figure 1) may be helpful. In such case, the state of the system evolves on
the ‘landscape’ to the nearest local minimum of the functional which
corresponds to the ‘landscape’.
 Sustainable and non-sustainable ways correspond to different minimum if
the bonds between elements are constant and internal patterns of elements
are fixed.
 But the change of norms, beliefs, ethical norms and concepts follows to the
deformation of ‘landscape’. So in this case the change of norms may push
the system from one minimum to other without increasing the ‘external
energy’ (functional of the system). Also, the changes in mentality can create
the new minimums in the system (create the sustainable way).

25. Sustainable development of
knowledge and of education
 The investigations on sustainable development have
been related mainly with the problems on natural
resources and energy.
 But only now it have been recognized that very
important (or just most important) became other
aspect of society life – namely KNOWLEDGE.
 The list of posing new problems and challenges are:
the development of knowledge; the knowledge use;
transfer of knowledge between generations;
spreading of knowledge; reproduction of knowledge
bearer (that is researchers, teachers, students) etc.

26. CONCLUSIONS
 In this talk we describe an approach to modelling social
systems and decision–making process in them which
can be useful for studying the sustainable development
in OR investigations.
 Also some frames for formalizing SD processes is
proposed
 The approach is based on the models, which have the
properties of associative memory and which allow to
incorporate the mental peculiarity of involved individuals.
 One such property – anticipation leads to the existence
of multi-valued solutions (which correspond to the
scenarios of system evolution). Even qualitative
consideration of such property allows understanding of
some aspects of decision–making processes.

27. REFERENCES
 Dubois Daniel, 1998. Introduction to computing Anticipatory
Systems. nternational Journal of Computing Anticipatory Systems,
(Liege), Vol. 2, pp.3-14.
 Haykin S., 1994. Neural Networks: Comprehensive Foundations.
MacMillan: N.Y.,
 Makarenko A., 1998. New Neuronet Models of Global Socio-
Economical Processes. In 'Gaming /Simulation for Policy
Development and Organisational Change' (J.Geurts, C.Joldersma,
E.Roelofs eds) , Tillburg University Press. 133- 138,
 Makarenko A., 2003. Sustainable Development and Risk Evaluation:
Challenges and Possible new Methodologies, In. Risk Science and
Sustainability: Science for Reduction of Risk and Sustainable
Development of Society, eds. T.Beer, A.Izmail- Zade, Kluwer AP,
Dordrecht, p. 87- 100.
 Zgurovsky M., Gvishiani A., 2008. Sustainable Development Global:
Simulation. Quality of Life and Security of the World Population
(2005 – 2007/ 2008). Kyiv: NTUU ‘KPI’, POLITECHNIKA. 336 p.