Markov Processes in Modeling Life Cycle of Economic Clusters 1 1 2 3 Galina Boush , Vitaly Shamis , Oksana Kulikova , and Svetlana Neiman 1 Omsk Humanities Academy, Omsk, Russia gboush@narod.ru, vitaliy1999@mail.ru 2 Siberian Automobile and Road Academy, Omsk, Russia ya.aaaaa11@yandex.ru 3 Omsk Institute of Service, Omsk, Russia svetlana1414@bk.ru Abstract. The object is economic clusters' life cycle (ECLC). The aim is to model its regulations algorithm. The hypothesis: ECLC transitions are stochas- tic processes independent of past. The method is Markov processes with dis- crete time. The algorithm is implemented in programming language Eсlipse. The results are: 1) mathematical model of ECLC with life-cycle stages, stochas- tic transitions, transition values. For mature clusters their borders and actors are defined; 2) ECLC modeling algorithm with given stochastic transition matrix considering influence of different environments; 3) software "Modeling _Cluster_v1" for Androids to model online cluster development based on ECLC regulations. The novelties: for the first time ECLC model considers stochastic character of life-cycle stages simulating the number of actors at each stage, cluster’s size, diameter and delimitation. Practical application: Program is ap- plied to education, for interest groups involved in cluster development and pro- jects to model of clusters’ evolution trajectories under different environmental changes. Keywords: stage in the life cycle, life cycle patterns, operational research, computer simulations of life cycle 1 Introduction The development of operational research in different subject areas seems to be quite relevant nowadays. One area where such methods are of great demand is the Science of Economics. This is due to the fact that improved management of the large-scale highly-sophisticated economic systems is associated today with the latest mathemati- cal tools since traditional economic methods used to solve the problems in the field demonstrate the tunnel vision and poor performance. One of the most promising economic systems of this kind is a cluster structure. However, despite the wide use of cluster approach to different areas and in different Copyright © by the paper's authors. Copying permitted for private and academic purposes. In: A. Kononov et al. (eds.): DOOR 2016, Vladivostok, Russia, published at http://ceur-ws.org 546 Galina Boush, Vitaly Shamis, Oksana Kulikova, and Svetlana Neiman countries, its comprehensive and detailed description and analysis in scientific litera- ture are absent; the processes of clusters’ formation and development remain poorly studied. As a consequence, it is difficult to consider and evaluate properly the devel- opment and management of cluster structures in the projects targeted for that. Howev- er, it is absolutely necessary in order for the clusters to ensure maximum generation of positive externalities. One of the most pressing issues in this regard is their life cycle. Further development of the theory of economic clusters and their practical im- plementation within the cluster approach requires mathematical and computer model- ing. The bibliographic survey reveals that in recent years some attempts have been made to simulate the life cycle [1, 2, 3, 4, 5, 6], degradation and collapse of cluster structures [7], [4], [8], etc. In these works nondeterministic model [3], Lotka-Volterra model [6], network theory [8] are used. However, as the analysis sums up, operational research methods have not been implemented yet to the study and modeling the selected aspects of the economic clus- ters, even though the methods have high heuristic perspectives. The considerations mentioned above, define the objective of the study to search for and apply the meth- ods of operational research, which allows accurate and adequate simulating economic clusters’ life cycle. 2 Theory and Methodology In the article the economic clusters are understood as non-institutionalized association of independent economic entities in a joint arrangement based on proximity (territori- al, sectorial, cultural), complementarity (product, resource, process), interconnectivity (material, nonmaterial, informational) [9, p. 162]; then the life cycle of the economic clusters is considered as the shift or change leading to transition in its basic state and pattern [9, p. 238]. Like any functional institutions, economic clusters have a life cycle, which represents a certain set and a sequence of stages, each of which is characterized by the special institutional structure and the status of the cluster. As a tool for simulating the life cycle of the cluster structure the Markov pro- cesses with discrete time are used. The choice of the method is determined by the fact that the life cycle of the economic clusters is a stochastic process independent of past. Hence, the discrete Markov processes with high confidence can describe the transition in the life-cycle stages. It should be noted that at present the Markov processes are applied to the stud- ies in various fields: in naval shipbuilding industry for planning and control the outfit- ting process in ships [10]; for industrial environments to predict the long-term evolu- tion of composting processes on an industrial scale [11]; for optical diagnostic pur- poses to simulate angular distribution of photons through turbid slabs [12]; in genetic studies for modeling the complex dynamics of mobile genetic elements within ge- nomes [13]; in chemistry to simulate interaction potential between particles in a col- loidal system and to control their assembly into a close-packed crystalline objects [14]; in cattle farming for unbiased prediction of the future performance of the ani- Markov Processes in Modeling Life Cycle of Economic Clusters 547 mals [15]; in building industry for cracking prediction on civil structures [16]; in med- ical science for optimization of anemia treatment in hemodialysis patients [17]; and even in music for enforcement of structure and repetition within music for bagana [18]. In economics according to the survey of the scientific literature the Markov processes are used to model optimal dynamic resource allocations between various companies to prevent defaults [19], to regulate social exchanges toward producing social equilibrium [20], to implement a social choice function [21], to maximize the expected discounted value of the total future profits [22], to solve optimal lending problem to certain types of borrowers [23], to examine the desirable sizes and policies of a strategic petroleum reserve for oil consumption countries [24], to show the opti- mality of a so-called save-up-to level policy and the existence of the optimal initial stock [25], to examine the financial optimality of disaster risk reduction measures [26] etc. But the Markov methods used to model clusters or in any other complex social- economic systems have not yet been found. 3 Model In the mathematical model of the economic clusters there are three types of actors: 1) Producers [products]; 2) Providers [resources]; 3) Consumers [products]. Each actor in the cluster implements the goal which determines its behavior in the market and is characterized by a number of indicators. 1) Producers constitute the basis of the market and are characterized by the two groups of indicators: a) indicators measuring the productive potential; b) indicators defining the rules of behavior. a) The group of the indicators measuring the productive potential of Producers includes: ─ innovative activity – discovering the potential of Producer for innovative activi- ty is determined by the method given in [27]. The range of values of the indicator is 0-40. The higher the value, the more easily Producer ventures into innovative markets; ─ internal capacity – , measuring the financial, manufacturing and human poten- tial of Producer is determined by the expert method based on the analysis of his fi- nancial conditions, productive and human capacities. Evaluation criteria: high, me- dium, low; ─ potential for adaptation and management – , reflecting the ability of Producer to adapt to changes in the environment is determined by the expert method on the ba- sis of the analysis of the management and the capacity to adapt to the external en- vironment. Evaluation criteria: high, medium, low. b) The group of indicators that define the rules of behavior of Producers is specified by the tuple or set of rules: (1) where – rules for acquiring all necessary resources for the manufacture of prod- ucts and their storage, including purchase price, volume, quality, warehousing system; – rules of production, simulating the production cycle and the quality of products; 548 Galina Boush, Vitaly Shamis, Oksana Kulikova, and Svetlana Neiman – rules of storage and realization of products that simulate warehousing of the finished products, cost and sales rules; – rules of behavior on the market, deter- mining the Producer’s goodwill, advertising policy, dealing with clients. 2) Providers supplying resources for Producers are the second group of actors in the economic cluster. Since the basic indicators for the analysis of interaction between Producers and Providers are the rules for acquiring resources included in the set of the Producers’ behavior rules, Providers in the model defines the indicators of the re- sources supplied. This is due to the fact that the relations of Producers and Consumers play the main part in the processes of cluster formation and development and, conse- quently, in implementing or patterning the life cycle of the economic clusters. The activities of Providers under these conditions are aimed at rapid offtake of the re- sources supplied. Providers are characterized by the following indicators: ─ price of a resource unit – in model monetary units; ─ amount of resource produced by one supplier – defined in model volumetric units; ─ quality of the resource – defined by the expert method based on the analysis of resources. Evaluation criteria: high, medium, low. 3) Consumers are characterized by the two groups of indicators: a) indicators deter- mining capacity; b) indicators defining the rules for inflow of funds and purchase of products. a) The group of the indicators determining the potential of Consumers includes: ─ amount of money that Consumer is willing to spend on the purchase of products – in model monetary units; ─ volume of purchased products – is defined as the ratio of available funds from Consumer to the value of the products. b) The group of indicators that define the rules of Consumer’s behavior is specified by the set of rules: (2) where – rules for the purchase of products on the market, including the Produc- er’s evaluation based on the rules of the group and the group , as well as the quality of the products manufactured (or sold); – funding rules. The economic clusters in the model discussed appear to be a system arranging the three interrelated components: production, resources and consumption. Each compo- nent is represented by the special type of actors. Production component serves as the core for the cluster system, determining the pro- cesses of cluster formation, the development of a uniform cluster structure or pattern and implementation of a certain life-cycle trajectory of a cluster [28]. The core of the economic cluster is a set of homogeneous enterprises producing goods with similar characteristics (Producers). The companies included in the cluster core form a com- pact set in (n)-dimensional space of characteristics. Once the core of the cluster is formed, it attracts Providers and Consumers. With it, between all the actors in the cluster the relations and ties of various levels of sustainability with the parameters and rules for cluster behavior are inevitably formed. Thus, the economic cluster can be selected when used the clustering methods of the (n)-dimensional space based on the analysis of distances between actors of the cluster [28]. Economic clusters are characterized by the following parameters: Markov Processes in Modeling Life Cycle of Economic Clusters 549 ─ availability of a cluster core, and relationships (informational, resource, communi- cative) between Producers; ─ the number of Producers in the cluster core structure; ─ the diameter of the cluster, which is defined on the basis of calculating the location of the cluster actors in space characteristics. The higher the level of development of the cluster, the smaller is the diameter; ─ the number of Providers in the cluster defined on the basis of the analysis of the conditions of enterprises entering the diameter of the cluster; ─ – the number of Consumers in the cluster, defined in the same way as the previous indicator; ─ the synergistic effect defined as the increase in production volumes with the in- crease of professional communication within the cluster without changing the pro- duction capacity of Producers. The economic clusters are self-organizing natural systems, in their life cycle they go through several stages: 1) diffuse group; 2) emerging cluster; 3) growing cluster; 4) mature cluster; 5) stagnant cluster. Shift in the economic life-cycle stages of a cluster is shown in Fig. 1. Fig. 1. Shift of economic life-cycle stages of cluster Characteristics of life-cycle stages of economic cluster are provided in Table 1. 550 Galina Boush, Vitaly Shamis, Oksana Kulikova, and Svetlana Neiman Table 1. Characteristics of life-cycle stages of economic cluster Emerging Growing Mature Stagnant Stage name Diffuse group cluster cluster cluster cluster Designation Existence of Cores Missing Available and Links Constant num- Missing Number of Produc- Missing, ber, depending ers in the Cluster representing Constant number on industry and Core diffuse group market Maximum size depending on Cluster Diameter - Decreasing Increasing industry and market Number of Provid- Constant ers in the Cluster Increasing de- Missing, pending on Number of Con- representing Increasing Decreasing industry and sumers in the Clus- diffuse group market ter Synergistic Effect Missing Available Missing Indicators characterizing Producers in economic cluster are listed in Table 2. Table 2. Indicators characterizing Producers in economic cluster Diffuse Growing Stagnant Stage name Emerging cluster Mature cluster group cluster cluster Designation Changing taking into Innovative account the specific Activity Changing Changing slight- activity of taking into each enter- ly or changing account Changing taking into account overall prise stochastically in specific trend of cluster development, vari- Internal each company, Changing activity of ance indicator reduced Potential variance indica- taking into each enter- Potential for tor increased account prise Adaptation specific and Manage- activity of each produc- ment er Clustering Clustering influ- Determined Clustering processes processes ence decreasing, Determined by market increasing, market greatly influ- market impact by market trends, influence decreas- encing, behav- Behavior of increasing, Pro- trends, weak weak rela- ing, behavior de- ior depending Producers ducers ' behavior relations tions be- pending on the in- on the interac- in the cluster between tween Pro- teraction between tion between becoming more Producers ducers Producers of cluster Producers of stochastic cluster Indicators characterizing Providers in economic cluster are given in Table 3. Markov Processes in Modeling Life Cycle of Economic Clusters 551 Table 3. Indicators characterizing Providers in economic cluster The emerging Stagnant Stage name Diffuse group Growing cluster Mature cluster cluster cluster Designation Resource price averaging and Resource Unit Depending on first cost and mar- depending on Changing slightly, de- Price ket resource acquisi- pending on market tion rules asked by providers Determined Resource by needs of Deter- Volume Pro- Determined by Determined by the capacity of Producers in mined by duced by one needs of Produc- market cluster and capacity ers in cluster Provider capacity of of market market Quality of re- Quality of resources averaging Quality of sources heteroge- and depending on rules of re- Not changing or slightly Resource neous, depending source acquisition, asked by changing on Provider Producer in cluster Consumers’ indicators in economic cluster are listed in Table 4. Table 4. Consumers’ indicators in economic cluster Diffuse Growing Mature Stage name Emerging cluster Key cluster group cluster cluster Designation Funding Depending on individual Consumer Volume of Purchased Depending on amount of funding of Consumer Products Formed to Averaging for Formed to Rules for Pur- Changing to meet individ- Consumers in clus- Similar to meet indi- chase of Prod- reflect influ- ual prefer- ter are developing Consum- vidual pref- ucts on Market ence of mar- ences and taking into influ- ers in erences and ket and com- wishes of ence of clustering cluster wishes of petition Consumer processes Consumer Funding Rules Individual Changing stages in the life cycle of the economic clusters operates in external envi- ronment. Classification of external environmental conditions is given in Table 5. 552 Galina Boush, Vitaly Shamis, Oksana Kulikova, and Svetlana Neiman Table 5. Classification of external environmental conditions of economic clusters External status economic cluster Condition Favorable Neutral Unfavorable A favorable polit- ical, economic, Little impact or no High or low level of competition Characteristics social and tech- impact of external on market, negatively affecting the of External nological condi- factors on the pro- activities of actors in the processes Conditions tions, optimal cesses of clustering of clustering market competi- tion Great probability of clustering Rapid clusters’ Gradual clusters’ Shift Trends in processes completion at early stag- development and development, the Cluster Life es of cluster development and state of growth possible existence Cycle Stages increase in period of stagnation in for a long time of early stagnation cluster As noted above, the life cycle of the economic clusters is a probabilistic stochastic process independent of past. This enables modeling and description of the transition processes in stages or phases of their cluster life cycle by means of the discrete Mar- kov process. Matrix of transitions between stages in the cluster life cycle is as fol- lows: (3) Calculations for each for each quantum of modeling time are performed according to the algorithms used for Markov process modeling [29]. The possible trajectories of life-cycle stage changes in economic clusters are calculated on the basis of the appro- priate calculating matrixes on the indices which have the maximum probability mean- ing if the particular cluster is studied at a certain life-cycle stage. There can be several such trajectories. After that, the indices of the number of the actors of the cluster and the change in its characteristics within the modeled time can be calculated. The transition matrix between life-cycle stages in the cluster is formed with the help of expert analysis and the results of the research in the functioning of the real economic clusters. For example, innovative economic clusters are developing more intensively and transforming into the stage of growth much more rapidly but as well they can be transforming into the stage of stagnation with the same high speed. It is due to the speed of the processes developing in them, the innovative ones being in- cluded. Agro-industrial clusters, for instance, are characterized by slower develop- ment as traditionalism and slow innovations are quite typical for them. Most of the Agro-industrial clusters in the world as well in Russia are mature clusters and they seldom transfer to the stagnation stage. Such regulations may be taken into account when building transition matrices between the life-cycle stages of economic clusters of various types. Markov Processes in Modeling Life Cycle of Economic Clusters 553 Calculation of the probability of economic cluster development at each stage of its life cycle can be performed for each quantum of modeling time considering specific environment. 4 An Algorithm of Modeling the Life Cycle of the Economic Clusters Modeling algorithm of the economic clusters’ life cycle includes the following stages: Stage 1. Setting a baseline of initial parameters for modeling life cycle of eco- nomic cluster, including the number of actors in each group (Producers, Providers, Consumers). Stage 2. Specifying the number of quanta or tacts of modeling time. Stage 3. Setting parameters for the life cycle of economic cluster. Stage 4. Specifying external environment for economic cluster. Stage 5. Developing a matrix of transitions between stages of the life cycle of economic cluster. For each condition of an external environment for the cluster the default range of the probabilities of transition from one life-cycle stage to another is specified. Developing matrices (2) is carried out on the basis of the random number generator considering the specified range of values. Stage 6. Visualization of the results of the Stage 5. If there is no need to adjust the matrix (2), you should go to the Stage 8, if it is necessary to correct the matrix (2), you should go to the Stage 7. Stage 7. User adjustment and correction of the matrix values (2). Stage 8. Calculation of the matrix values (2) for each quantum of modeling time. Stage 9. Visualization of calculation in the Stage 8. Stage 10. The choice of the most probable cluster life-cycle stages for each quantum of modeling time. The selection is done by ranking the values consider- ing the specified range of values. If all values miss the specified range, you should go to the Stage 5, if not – you should go to Stage 11. Stage 11. Visualization of calculation of the Stage 10. Stage 12. Delimitation of the cluster and cluster actors for each quantum of modeling time using algorithm FOREL [30]. Stage 13. Calculation of the life cycle of economic cluster for each quantum of modeling time. Stage 14. Calculation of indicators for all actors in the economic cluster, and for each quantum of modeling time. In the transition from the first to the subsequent stages of the economic cluster life cycle averaging the cluster indicators for the cluster actors takes place by reducing the confidence interval and, vice versa, when you re- turn from the subsequent stages of the life cycle of the cluster to the previous ones, there is an increase in the dispersion of values or in the variance indicator for each of the indicators of cluster actors. Also in the transition from one stage of the economic cluster life cycle to another the number of cluster actors can vary. Stage 15. Cognitive visualization of results of the calculations using the Dash- board technology [31]. 554 Galina Boush, Vitaly Shamis, Oksana Kulikova, and Svetlana Neiman 5 The Simulation Program of the Life Cycle of the Economic Clusters Based on the algorithm of simulating the economic clusters’ life-cycle the Program "Modeling_Cluster_v1" for Android-based devices is developed. The choice of the Android operating system is due to the independence of this device from hardware and to its performance characteristics as well. In Fig. 2 a screenshot of the main window of the Program is given. Fig. 2. Main Window of Program "Modeling_Cluster_v1" 6 Conclusions The following results were obtained in the article presented, namely: 1. mathematical model of life cycle of the economic clusters including the life-cycle stages and transitions between them; probability values of transitions. For stages with a mature economic cluster, the problem of determining the boundary or de- limitation parameters of the cluster is solved, its actors are defined, changing in its number at different stages of the life cycle is considered; 2. modeling algorithm of the life cycle of the economic clusters with given transition probability matrices between the stages of the cluster life cycle, for the impacts of different types of environment can be specified. The algorithm allows calculating the most probable trajectory of the life cycle of the economic clusters, as well as determining their sizes and actors at each stage; 3. software product "Modeling_Cluster_v1" implemented for Android-based devices which allows online management of the cluster development on the basis of the identified patterns of transition between the stages of the economic clusters life cy- cle to implement modeling its evolution taking into account environment impact. Markov Processes in Modeling Life Cycle of Economic Clusters 555 Simulation results are presented in the form of Dashboard, visualizing the changes of stages in the life cycle of economic clusters and the behavior of its actors at each stage. Thus, such methods as the Markov processes with discrete time provide mod- eling the life cycle of the economic clusters. This made possible to identify the regula- tions in the development of the specific economic clusters and determine both their sizes and characteristics of their actors at each stage of the life cycle. The novelty of the results obtained is that for the first time the model of the economic cluster life cycle takes into account the probability character of transitions between the stages, allows model the number of actors participating in each of the stages, the size, the diameter and the delimitation of the cluster. The software product may be used for the educational purposes, as well as for executive bodies and interest groups involved in the development and implementation of cluster programs and projects, also for modeling possible trajectories of economic cluster evolution in changing economic environments. References 1. Ratner, S.V., Akinkina, M.M.: Vyibor parametrov optimalnogo upravlencheskogo vozdeystviya na regionalnyiy neftegazovyiy klaster na osnove imitatsionnogo modelirovaniya. Regionalnaya ekonomika: teoriya i praktika. 20, 2–11 (2011) [Ратнер, С.В., Акинкина, М.М.: Выбор параметров оптимального управлвенческого воздейст- вия на региональный нефтегазовый кластер на основе имитационного моделирования. Региональная экономика: теория и практика. 20, 2–11 (2011)] 2. Smirnova, S.M.: Modelirovanie stadii razvitiya promyishlennogo klastera. Nauchnoe obozrenie. 8, 159–162 (2013) [Смирнова, С.М.: Моделирование стадии развития про- мышленного кластера. Научное обозрение. 8, 159–162 (2013)] 3. Popp, A., Wilson, J.: Life cycles, contingency, and agency: Growth, development, and change in English industrial districts and clusters. Environment and Planning A. 39 (12), 2975–2992 (2007) 4. Press, K.: Divide to conquer? Limits to the adaptability of disintegrated, flexible specializa- tion clusters. J. of Economic Geography. 8 (4), 565–580 (2008) 5. Suire, R., Vicente, J.: Clusters for life or life cycles of clusters: in search of the critical fac- tors of clusters’ resilience. Entrepreneurship and Regional Development. 26 (1–2), 142–164 (2014) 6. Tsai, B.-H., Li, Y.: Cluster evolution of IC industry from Taiwan to China. Technological Forecasting and Social Change. 76 (8), 1092–1104 (2009) 7. Bek, M.A., Bek, N.N., Sheresheva, M.Y., Johnston, W.J.: Perspectives of SME innovation clusters development in Russia. J. of Business and Industrial Marketing. 28 (3), 240–259 (2013) 8. Zeng, Y., Xiao, R.: Modelling of cluster supply network with cascading failure spread and its vulnerability analysis. International J. of Production Research. 52 (23), 6938–6953 (2014) 9. Boush, G.D.: Klasteryi v ekonomike: nauchnaya teoriya, metodologiya issledovaniya, kont- septsiya upravleniya. Omskiy gosudarstvennyiy universitet, Omsk (2013) [Боуш, Г.Д.: Кластеры в экономике: научная теория, методология исследования, концепция управ- ления. Омский государственный университет, Омск (2013)] 556 Galina Boush, Vitaly Shamis, Oksana Kulikova, and Svetlana Neiman 10. Dong, F., Deglise-Hawkinson, J.R., Oyen, M.P., van, Singer, D.J.: Dynamic control of a closed two-stage queueing network for outfitting process in shipbuilding. Computers & Op- erations Research. 72, 1–11 (2016) 11. Fernández, C., Mateu, C., Moral, R., Sole-Mauri, F.: A predictor model for the compost- ing process on an industrial scale based on Markov processes. Environmental Modelling & Software. 79, 156–166 (2016) 12. Li, X., Northrop, W.F.: A Markov Chain-based quantitative study of angular distribution of photons through turbid slabs via isotropic light scattering. Computer Physics Communica- tions. 201, 77–84 (2016) 13. Drakos, N.E., Wahl, L.M.: Extinction probabilities and stationary distributions of mobile genetic elements in prokaryotes: The birth-death-diversification model. Theoretical Popula- tion Biology. 106, 22–31 (2015) 14. Bevan, M.A., Ford, D.M., Grover, M.A., Shapiro, B., Maroudas, D., Yang, Y., Thyagarajan, R., Tang, X., Sehgal, R.M.: Controlling assembly of colloidal particles into structured ob- jects: Basic strategy and a case study. J. of Process Control. 27, 64–75 (2015) 15. Kristensen, A.R.: From biological models to economic optimization. Preventive Veterinary Medicine. 118 (2–3), 226–237 (2015) 16. Kobayashi, K., Kaito, K., Lethanh, N.: A competing Markov model for cracking prediction on civil structures. Transportation Research Part B: Methodological. 68, 345–362 (2014) 17. Escandell-Montero, P., Chermisi, M., Martínez-Martínez, J.M., Gómez-Sanchis, J., Barbieri, C., Soria-Olivas, E., Mari, F., Vila-Francés, J., Stopper, A., Gatti, E., Martín-Guerrero, J.D.: Optimization of anemia treatment in hemodialysis patients via reinforcement learning. Arti- ficial Intelligence in Medicine. 62 (1), 47–60 (2014) 18. Herremans, D., Weisser, S., Sörensen, K., Conklin, D.: Generating structured music for bagana using quality metrics based on Markov models. Expert Systems with Applications. 42 (21), 7424–7435 (2015) 19. Ayesta, U., Erausquin, M., Ferreira, E., Jacko, P.: Optimal dynamic resource allocation to prevent defaults. Operations Research Letters. 44 (4), 451–456 (2016) 20. Dimuro, G.P., Costa, A.C., R., da: Regulating social exchanges in open MAS: The problem of reciprocal conversions between POMDPs and HMMs. Information Sciences. 323, 16–33 (2015) 21. Renou, L., Tomala, T.: Approximate implementation in Markovian environments. J. of Eco- nomic Theory. 159 (A), 401–442 (2015) 22. Ben-Zvi, T., Chernonog, T., Avinadav, T.: A two-state partially observa- ble Markov decision process with three actions. European J. of Operational Research. 254 (3), 957–967 (2016) 23. So, M.C., Thomas, L.C., Huang, B.: Lending decisions with limits on capital available: The polygamous marriage problem. European J. of Operational Research. 249 (2), 407–416 (2016) 24. Bai, Y., Zhou, P., Zhou, D.Q., Meng, F.Y., Ju, K.Y.: Desirable policies of a strategic petro- leum reserve in coping with disruption risk: A Markov decision process approach. Comput- ers & Operations Research. 66, 58–66 (2016) 25. Wen, X., Xu, C., Hu, Q.: Dynamic capacity management with uncertain demand and dy- namic price. International J. of Production Economics. 175, 121–131 (2016) 26. Espada, Jr., R., Apan, A., McDougall, K.: Spatial modelling of natural disaster risk reduc- tion policies with Markov decision processes. Applied Geography. 53, 284–298 (2014) 27. Balashov, A.I., Rogova, E.M., Tkachenko, E.A.: Innovatsionnaya aktivnost rossiyskih predpriyatiy: problemyi izmeneniya i usloviya rosta. Sankt-Peterburgskiy gosudarstvennyiy politehnicheskiy universitet, Sankt-Peterburg (2010) [Балашов, А.И, Рогова, Е.М., Тка- ченко Е.А.: Инновационная активность российских предприятий: проблемы измере- ния и условия роста. Санкт-Петербургский государственный политехнический уни- верситет, Санкт-Петербург (2010)] Markov Processes in Modeling Life Cycle of Economic Clusters 557 28. Boush, G.D., Kulikova, O.M., Shelkov, I.K.: Agentnoe modelirovanie protsessov klasteroobrazovaniya v regionalnyih ekonomicheskih sistemah. Ekonomika regiona. 1, 64– 77 (2016) [Боуш, Г.Д., Куликова О.М., Шелков, И.К.: Агентное моделирование про- цессов кластерообразования в региональных экономических системах. Экономика ре- гиона. 1, 64–77 (2016)] 29. Daniel, W.: Stroock An Introduction to Markov Processes (Graduate Texts in Mathematics). Springer Berlin Heidelberg (2005) 30. Mashinnoe obuchenie. Raspoznavanie, http://www.machinelearning.ru 31. Infographer, http://www.infographer.ru/dashboard_practice