Functional Elements (Business Processes)

The process of reengineering the functional elements of management information system must presented as stable, in which all the kernel functional components of the management information system should be saved. Problem statements: There is no formalized models for stabile reengineering for the functional structure (functional elements) for various information systems types, therefore the process of reengineering is not stabile and may be it finished with failure reengineering process or failure information system. Approach: The information system development Approach and the mathematical methods to formalized the detailed mathematical model for the information systems stabile reengineering, and at different stages of IS reengineering. Results: All process of stabile reengineering at the various stages of it was formalized in mathematical models. Conclusion: The developed models helps the system analyst to define if the reengineering process for the functional structure of information system will be stabile and effective or not, and if it more feasible than creating new information system.


INTRODUCTION
Now majority of existing Information Systems (IS) are represented by complicated systems, which with the current of time developed, improved, or upgraded in the correspondence with changed requests of the enterprisecustomer. In this case the perfecting IS usually happens within the framework of the existing standards, techniques and methods of operation of business, to which concern: Standards MRP (Management Resource Planning) and MRPII, Concept CRP (Client Resource Planning), Supply Chain Management (SCM), Concept ERP (Enterprise Resource Planning) and Enterprise Resource and Relationship Processing/Enterprise Commerce Management (ERPII/ECM) [1] .
Reengineering should be an important goal of any effort to develop an Management Information Systems (MIS) plan for an organization symptoms of obsolete design concepts are manifested throughout almost any large application developed a decade or more ago in its failure to satisfy cross functional needs, its limited functional capabilities, its inability to provide selected and tailored information drawn from widespread parts of the organization, its flexibility in meeting changing needs, and its unfriendly interfaces. Traditionally the argument has always been made that the IS planning should closely follow the organizational strategic planning process. The intention is to eventually align the two plans such that execution of strategic plans can be closely supported by organizational information systems. The rationale is that all levels of decision making require ready access to information and if the two plans are well-aligned it would result in development of systems that are highly effective in making decisions [2,3] .
However, experience has shown that most organizational systems are developed with no attention to the needs for cross-functional business processes that play an important role in the proper functioning of the organization. These business processes are defined as a collection of activities that takes one or more kinds of input and creates an output that is of value to the customer. Alternatively, business processes can also be defined as, "a set of logically related tasks performed to achieve a defined business outcome." Processes can be understood as "beginning and end points, interfaces, and organization units involved, particularly the customer unit." for Examples of such processes include processing an insurance claim, sales and delivery, management of supply chains, and so on. These business processes become the focal point of every reengineering effort and entail the need to go beyond traditional tasks and functions [4,5] .
However, despite more and more innovative technological solutions for business processes, resistance to change continues to be the biggest BPR obstacle. According to many experts, this is the main reason why reengineering projects in many organizations fail [6][7][8] .
Problem statement: However all listed above standards, techniques and methods are oriented, first, to develop new IS or new subsystems. At the same time, the main problem of creation IS consists mainly of modernizing, development and integration of existing information systems of various subordination in uniform IS of the enterprise or association of the enterprises. In this case the processes of integration limited by the purposes, generally delivered before the enterprise; by necessity of simultaneous mutual coexistence and operation of separate solutions within the framework of uniform IS, by availability of the certain amount of resources (safe life), which ensure operation and development IS.

MATERIALS AND METHODS
The development of Information Systems (IS), information system reengineering and the mathematical models of IS at various stages of reengineering [9][10][11][12] . May be proposes to classify all the operations performed at different stages of IS reengineering into two classes: • Extensive ones, directed on increase of the number of various elements added to IS (new functional tasks and modules, software and technical complexes, database tables, queries to databases • Intensive ones that allow improving the IS structures according to the changes in external and internal condition of functioning Operations of the both classes should be performed in compliance to the following requirements: • IS correspondence to new requirements and criteria determined during the reengineering • Stability of 'the kernel' (main elements) of IS to changes that occur during IS reengineering Thus IS reengineering regardless of class of the operations should be considered as solution of the problem of determination of IS state of the totally of states which are the most stabile under the given conditions of exploitation as well as correspond to the given efficiency criteria and requirements to IS [13][14][15][16][17] .
General model information systems: Purpose we used the complex IS mathematical models based on the mathematical apparatus of the general theory of systems. According to this theory IS should be described with the aid of the following model [9] :

General model of reengineering information system:
Then the conditions of IS during reengineering can be described as follows: Suppose that during IS exploitation some defects caused by lapses in functioning of business-processes were found. To eliminate these defects we need to change the IS functional structure L Fu its model in a general case coincides with the model (1) and looks as follows: Here: Whatever modifications should be made in the supporting structure of IS, the model of functional structure L Fu after the reengineering processes will be as follows: Here: That takes into consideration the proposed solutions concerning the functional structure L Fu : Here: { } It can hardly seriously affect the developed above model Fu S′ that can be described using conditions: Fulfillment of conditions (11)(12)(13) determines the structural stability of IS to changes and thus ensures meeting the second requirement to IS reengineering.
Stabile IS reengineering can be presented as follows. IS structure (functional or structure of support type) passes into a new stabile state, if the number of changes in IS during reengineering is minimal. In this case the sum of values of efficiency criteria for IS functioning before the reengineering must be less or equal to the sum of values of efficiency criteria for IS functioning after the reengineering. Suppose Suppose f 1 is a number of problems that can be solved in the IS, f 2 -amount of the transacted information; f 3time of transaction, f 4 -of queries to databases, f 5 -right of access to the data. Suppose for the selected f i : here F ∆ is difference between vectors of requirements to the system. If F 0 ∆ = then there is no necessity in reengineering, otherwise F 0 ∆ ≠ -the system must be reengineered.
The process of IS reengineering, which is a transition of IS from the actual state (the state of IS during its industrial use) to other, desired state (a state determined by new requirements of users, new features of business-processes, can be described as follows: Then taking into account the modifications made during reengineering the structure of IS can be described as follows: Here i jL C are specific expenses on reengineering.
The described models of IS support types' structures and of IS functional structure are metamodels used for determination of general rules of transformation of requirements and substitutions for reengineering of IS support types. In order to detail these models we proposes to describe 'primary elements' of IS support types with the help of local technical and technical-economical criteria of efficiency analyzed in the paper [18] . Then fulfillment of the first requirement of stabile reengineering of a particular type of IS support can be described using the following reengineering function:  In accordance to the proposed formalized descriptions we proposes to consider the task of stabile IS reengineering as the problem of determination such a state or the totality of states of IS support types' structures that in a general case can be described as following: Here * ω(S ) is a description of IS using metamodels similar to (1) after completion of reengineering.

RESULTS
The article contains a theoretical generalization of and a new solution of the following scientific problems: Development of a complex of IS mathematical models at different stages of reengineering; development of a mathematical model of stabile IS reengineering; development of formalized description of reengineering function and structural stability function. Using these models and descriptions, we developed the "primary elements" of technology stable reengineering IS. Its usage at different stages of IS designing allows to improve the efficiency of IS modernization and to reduce the amount of work on modification of individual supports types. The results of the analysis of main approaches to enterprise and IS reengineering as well as of existing methods and instrumental tools of enterprise and IS reengineering lead us to the conclusion that the main drawback of the existing methods and tools of IS reengineering is lack of a single complex of interrelated models and methods of analysis and synthesis of structures of IS support types optimal by the given efficiency criteria of structures of IS support types. This determines the main scientific purpose of the article and the tasks of the study.

DISCUSSION
Stabile reengineering mathematical models: The process of stabile reengineering at the various stages of it was formalized in mathematical model. The models are support varicose types of information systems. The function (26) can be used to complete the stabile process of reengineering Information-Retrieval IS (IRIS), the description in of structural stability function (27), exactly used to complete the process of reengineering Information-Analytical IS (IAIS) and the description of structural stability function (26) is used for support types of Information-Controlling IS (ICIS). The generally description of structural stability function which presents the problem of determination such a state or the totality of states of IS support types' structures can be described in description (29). All these mathematical descriptions can de used to evaluate the task of reengineering information systems at the various their types, and determine if the process of reengineering is feasible or not without starting the process of reengineering with real projects. This is the first time when process of reengineering and the process of stabile reengineering will be formalized in mathematical functions. Every reengineering project manager can use these models to define the success of the project early.

CONCLUSION
The developed models helps the system analyst to define if the reengineering process for the functional structure of information system will be stabile and effective or not, and if it more feasible than creating new information system.