Mathematical model and coordination algorithms for ensuring complex security of an organization

The mathematical model of coordination when ensuring complex security of the organization is considered. On the basis of use of a method of casual search three types of algorithms of effective coordination adequate to mismatch level concerning security are developed: a coordination algorithm at domination of instructions of the coordinator; a coordination algorithm at domination of decisions of performers; a coordination algorithm at parity of interests of the coordinator and performers. Assessment of convergence of the algorithms considered above it was made by carrying out a computing experiment. The described algorithms of coordination have property of convergence in the sense stated above. And, the following regularity is revealed: than more simply in the structural relation the algorithm, for the smaller number of iterations is provided to those its convergence.


Introduction
Security of the organization represents the complex category including at least the following local components: providing mode and protection, fire security, information security, health and security of people. Need of coordination arises owing to the fact that the persons making private decisions on ensuring separate aspects of security are allocated with a certain independence. Not only the exception, but infringement of this independence reduces security as is followed by removal of responsibility from subordinates when performing of the functional duties by them. At the same time, the independence leads to the conflict of interests "private-part" and "private-general" which permission requires adoption of the complex coordinated decision considering both local interests, and interests at the level of all system in general [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15].          ... ,

Formulation of a problem
where 14 14 , ..., , , The obvious shortcoming (1) is that she has no constructive decisions as in neyaspekta of management and coordination are set implicitly. Directly to mark out the specified aspects we detail and we concretize (1). For this purpose, we will consider the control system of complex security of the organization represented on the scheme figure 1. On this scheme are designated by symbols: С0 -the coordinator who is responsible for all system of security of the organization in general and coordinating work of performers; С1, С2, С3, С4performers, the exercising direct control of processes of ensuring local of the organization; Р1, Р2, Р3, about the current mismatch of private processes of ensuring local security Р1, Р2, Р3, Р4, seeks to minimize a deviation of all process P from the required (reference) state that in an assumption of onedimensionality of space   r , that is when   rr  , formally is expressed as the following criterion function: the rated expert coefficients reflecting the importance of indicators 14 q ,...,q in the general structure of security of an object which contents reveals in table 1.
Thus, the substantial party of mathematical model of coordination at adoption of the complex decision on security of the organization to be in that the coordinator on the basis of information on the level of mismatch of private decisions ij  it developed and has brought such coordinating influences to subordinates i y (i 1,4)  , which will help them to develop the managements of the security measures providing the greatest possible integrated effect at the level of the organization, but in too time keeping "optimality" of management of own processes from the point of view of local interests.

Assessment of mismatch of private processes of security of the organization
The theoretical base of such assessment is made by the typology of interference of private processes of ensuring local security of Р1, Р2, Р3, Р4 presented in table 2 [1,5]. Table 2. Types of paired interaction between the private security processes of the institution and the corresponding level of inconsistency assessment.
Size  defines type and the maintenance of an algorithm of coordination [3,4]. If 1 0.33     , then in system essential mismatches take place, and it is necessary for effective coordination that instructions of the coordinator dominated over decisions of performers. At 0.33 1    in system there are practically no mismatches concerning security, and effective coordination can be reached if the solution of problems is given not "payoff" to performers. If 0.33 0.33     , then in system insignificant mismatches on minor questions which can be eliminated on the basis of parity of interests of the coordinator and performers [5,[16][17] take place.

Coordination algorithm at domination of instructions of the coordinator
In this case the criterion of efficiency of coordination has to be based on comparison of sizes r and r , reflecting the flowing and demanded levels of security of all system in general. Taking into account it the algorithm of coordination comes down to the step-by-step iterative procedure of the following look.
Step 1. Using a program method of generation of random numbers, we set a starting vector of the coordinating influences (0) i y Y;(i 1,4) . Random numbers have to get out of an interval from 0 to 1, at the same time their sequence has to submit to the uniform law of distribution.
Step 2. By method of calculus of variations, we solve problems (3). Also, we remember Step 3. For the received values we solve a problem (2) and we remember values and (0) r .
Step 4. We compare (0) r to the required level of complex security   r . If (0) rr  the task is solved, the coordinating influences (0) i y (i 1,4)   are recognized as optimum. As (0) rr   we continue the solution of a task.
Step 8. We compare (1) r to the required level of complex security   r . If (1) rr  the task is solved, the coordinating influences (1) i y (i 1,4)   are recognized as optimum. As (0) rr   we continue the solution of a task.
Otherwise  

Coordination algorithm at parity of interests of the coordinator and performers
Unlike two previous cases, the criterion of efficiency of coordination has to be based as on comparison of sizes r and r , reflecting the flowing and demanded levels of security of all system in general, and on comparison of sizes   i Q i 1, 4  and i Q , reflecting the flowing and demanded levels of separate aspects of security. Taking into account it the coordination algorithm at parity of interests of the parties comes down to the step-by-step iterative procedure of the following look.
Step 1. Using a program method of generation of random numbers, we set a starting vector of the coordinating influences (0) i y Y;(i 1,4) .
Step 2. One of methods of calculus of variations we solve problems (3). The received values Step 3. For the received values we solve a problem (2)  are recognized as effective. If at least one their specified conditions aren't satisfied, then we continue the solution of a task.
Step6. We solve problems (3) rr  the attempt  is considered unsuccessful. Search stops after unsuccessful steps.

Conclusion
Assessment of convergence of the algorithms considered above it was made by carrying out a computing experiment. At the same time, it was necessary that the algorithm meets if for the number of iterative cycles (about 150-200) accepted for practice, it allows to receive established results, that is results having no more than 10% a deviation from average value at tenfold repetition of cycles. Results of a computing experiment are presented in table 3. From the analysis of the obtained data it is visible that the described algorithms of coordination have property of convergence in the sense stated above. And, the following regularity is revealed: than more simply in the structural relation the algorithm, for the smaller number of iterations is provided to those its convergence. So, for example, for the simplest algorithm of coordination at domination of interests of performers the convergence is reached approximately for 80-90 cycles, in case of the most difficult a coordination algorithm at parity of interests of the coordinator and performers -for 120-160 cycles.