INCENTIVE SYSTEMS IN PROJECT MANAGEMENT

Analysis of the current state of project management shows a significant increase in the number of projects implemented in the spheres of construction, public administration, energy, consulting, and digital technologies. Professional project management allows to effectively allocate responsibilities and responsibilities between project participants, save money and time, reduce risks. Understanding the importance of future development, the heads of organizations are increasingly using project management methods to create competitive advantages, and in the applied theory of management of organizational systems, more attention is paid to the teamwork of the organization's personnel. An important task in the management of the project team is the formation of a system of incen-tives for its participants, whose task is to ensure the implementation of the project on time without additional budget expenditures. In order to maintain the motivation of project team members, individual and unified incentive schemes. The disadvantage of individual incentive systems is, firstly, unequal conditions for encouraging different groups of performers, and secondly, the danger of manipulation, that is, deliberate distortion of information on the amount of work and costs transferred from the members of the project team. In order to exclude distortion of information and reduce tension in the project team, unified incentive schemes apply when the incentive mechanism is the same for all performers. Intermediate position is occupied by the group incentive schemes, when a number of project operations are divided into groups, for executors of each of which a unified incentive system is applied. The article considers jumping unified stimulation systems, and the problem of choosing the optimal group incentive system is solved.


Introduction
reflected in the works of B.N. Burkov [3], D.A. Novikova [4,5], A.G. Chkhartishvili, V. Holmstrom, J. Marshak, R. Radner. The current state of mathematical models of the formation and functioning of teams includes several research directions: assignment models that use, in the main, the optimization apparatus for solving the tasks of forming the composition of teams, the distribution of roles and the scope of work; game-theoretic models; simulation models; Reflexive models using the apparatus of the theory of reflexive games.
An important task in the management of project teams is the formation of incentive systems for project team members. In this case, it is necessary to take into account the specifics of the project activity [1,2]. To the conditions of the project it is unproductive to apply methods of stimulating current work. The task of the incentive system in project management is to ensure that the project is completed on time without additional budget expenditures. The calculation of incentive payments should be based on not only individual, but also team performance indicators.
When creating incentive systems for project team members, it is necessary to take into account the specifics of the project activity. To the conditions of the project it is unproductive to apply methods of stimulating current work. The task of the incentive system in project management is to ensure that the project is completed on time without additional budget expenditures. The calculation of incentive payments should be based on not only individual, but also team performance indicators.
In order to maintain the motivation of project team members, individual and unified incentive schemes are applied. The disadvantage of individual incentive schemes is, firstly, unequal conditions for encouraging different groups of performers, which can cause tension in the project team. Secondly, there is a danger of manipulation, that is, deliberate distortion of information on the amount of work and costs transferred from the members of the project team. In order to exclude distortion of information and reduce tension in the project team, unified incentive schemes apply when the incentive mechanism is the same for all performers. Intermediate position is occupied by group incentive schemes, when a number of project operations are divided into groups, for executors of each of which a unified incentive system is applied.

Formulation of the problem
Linear incentive systems use the linear function of the time of reduction of operations i i h   , where  -the standard for reducing the duration, which is uniform for all operations, i  -is the planned decrease in the duration of the i-th operations. In the spasmodic incentive system, the premium С to the performers of the operation does not depend on the magnitude of the reduction of the operation i  and is the same for all transactions for which 0 i   . We consider linear cost functions of the form ( ) , 1, . Let the project be a sequential chain of n operations. Let's assume that the operations are numbered in ascending order i k , i. e. 1 2 ... n k k k    . The value i D defines the standard duration of the i-th operation, and the value i d -the minimum possible. It is necessary to build an optimal incentive system. For the construction of an optimal individual incentive system, we determine such a number j, that is Determine the total amount of premiums. The amount of bonuses for operations 1, The amounts of bonuses for We construct an optimal unified incentive system. Theorem 1. There is a unified system, such that the planned duration of operations for individual and unified incentive systems are the same, the standard j k   , and the total amount of premiums is j k  .
Evidence. Given a standard λ the reduction in duration is possible for operations for which j k   , that is, for operations for which the premium is not negative. It follows from condition (1) (2).
The theorem is proved.
The difference between the total premiums of the unified and individual incentive systems is

Jumping Unified Incentive Systems
Jumping unified incentive systems are characterized by the fact that premium С is the same for all operations for which ij Let the premium value С be given. In this case, the task is to shorten the duration of the project to the required value so that the number of operations with 0 ij   s minimal. We show that this problem is reduced to a discrete problem of optimizing the network by cost. For a given value of С the maximum reduction in the duration of the operation (i, j) is equal to min ; .
Thus, the duration of each operation can take two values: the minimum value D ij with zero costs (premium) and the minimum value with costs (premium) С for any operation. A task. Determine the duration of operations so that the duration of the project was no more (Т -θ), and the costs were minimal.
Discrete optimization problems belong to the class of complex problems of discrete optimization. Consider the solution of this problem for aggregated networks. For this it is sufficient to consider a solution for a serial and parallel chain.
Decision. Order the operations in descending order η i . We define the minimal number of operations such that the sum η i is at least θ.
For independent (parallel) operations, the problem is solved more simply. Solving problems for sequential and parallel operations, we get the dependence S(T) of the premium value from Т for the whole aggregate project. Example 1. Consider the network of Fig. 1  We will be limited to consideration of C-systems. 1 step. We consider sequential operations (0, 2), (2,3). We have a dependence ( ) The described algorithm can be used to obtain lower bounds, transforming an arbitrary network into an aggregate.

Group incentive schemes
Consider the problem of choosing the optimal group incentive system in the following formulation. A plan was drawn up to shorten the duration of the project, in which the duration and operations are reduced by Δ i having the cost function coefficients K i with a linear incentive system, the incentive rate is , and the incentive fund is We will set the task of synthesizing the optimal group incentive system for a given number of groups m n  (it is clear that when we have an individual incentive system m n  ). Let the norms be defined  Evidence. Let there be an operation S such that K s ≤ λ j , but 1 S Q  , where l > j. The award to the performers of the operation S is equal to l S    . We transfer the operation S to the group j. In this case, the premium will be j S    , less than l S    . The theorem is proved.

Fig. 2
Let K i be numbered in ascending order, that is, The arc i, j of the graph reflects the fact that the operations i + 1, т, j belong to one stimulus group, and the incentive standard in this group is K j . Note that any path in a graph joining an input with an output and having т arcs determines uniquely the partition of operations into groups and the corresponding norms λ j . The reverse is also true, to any partition of the executors of operations in the team into incentive groups there corresponds a certain path in the graph connecting the entrance with the exit.
Define the arc length (i, j) We show that l ij is equal to the fund for stimulating the group of operations (i + 1, m, j). Indeed, the incentive standard for this group is K j , and the total reduction in the durations of group Consequently, the length of any path from the т arcs connecting the entrance to the exit is equal to the value of the incentive fund.
Task. Define т -is the path (the path containing т arcs) of minimal length. This problem is a rather complex task of discrete optimization. Taking into account that the number of groups is usually small, the problem can be solved by brute force.
A similar problem can be solved for the C-stimulus system. Let the operations, the reduction values of their durations, and the coefficients K i be given. The size of the premium is obviously equal to max i i С K   and the incentive fund Ф C n   . Let all operations be divided into т groups and numbered in ascending order i i K  , i. e.
We denote by j Q the set of operations for which i i j K C   . Then it is obvious that The problem, like the previous one, reduces to determining the shortest m-paths in a graph with arc lengths With a small number of groups, the problem, like the previous one, can be solved by exhaustive search.

Conclusion
Group incentive systems occupy an intermediate place between individual systems and unified systems. They to some extent reduce the shortcomings of individual systems, as they generate competition both within groups and between groups and at the same time, as a rule, more effective than unified ones. The problem of constructing optimal group systems for arbitrary network graphs, and the development of effective algorithms for determining shortest paths with a fixed number of arcs are of interest.