The paper deals with network programming method and its application for solving multi-extreme and discrete optimization problems in Project Management. The purpose of the paper is to develop a new approach to solving optimization problems, called the network programming method and to solve a number of optimization problems of project management based on this method. This method is used to solve the problem of building a business-supporting project portfolio. The life cycle of a building project can be represented as a great number of interrelated and interacting actions leading to the achievement of the set goals in the presence of existing constraints (resources, time, cost, quality, reliability, safety, etc.). Such a representation is called a network representation. On the basis of the network programming method and its special case - the dichotomous programming method, new algorithms for solving a number of optimization problems in project management are proposed. The proposed method makes it possible to use it in solving the problems of managing the life cycle of a construction object. It can be used not only throughout the entire life cycle, but also independently during the planning, implementation and completion phases of project management.

Using the example of the steelmaking production of JSC EVRAZ United West Siberian Metallurgical Plant, the paper considers the task of synchronous calendar planning in the interval of several planned periods of operation of basic oxygen furnace (BOF), BOF shops, production as a whole, as well as ongoing repairs of BOF for steelmaking production (two BOF shops with two and three BOFs). Scheduled stops of the BOF for repair depend on the actual achieved duration of the lining campaign and production schedules of the units and are performed when the current duration of the BOF campaign reaches a given standard value. Thus, the current duration of the BOF campaign is described by a discrete, nonlinear quasi-periodic function that does not have a fixed period, but has some regularity. Technological limitations were forma­lized, determining the minimum and maximum values of the number of melts per day that each of the workshops can produce with one or two BOFs operating simultaneously. The authors formulated the conditions to avoid performing two “cold” repairs in one shop in one planned period and ensuring daily processing by BOF shops of all cast iron coming from the blast furnace shop. In the proposed mathematical formulation of the problem, it is required to find such schedules of BOF repairs and such calendar plans of their work that satisfy the formulated constraints and optimize the non-linear criterion. The proposed criterion is aimed at ensuring the constant readiness of the shops for implementation of the production program and design productivity. The task is formulated for the conditions of trouble-free operation and stable provision of the shops with liquid cast iron as the main component of the metal charge of BOF smelting.

Improvement of management mechanisms of making and scheduling development programs is an important area of performance improvement (achievement of formulated targets) and operation efficiency (reduction of the amount of consumed resources) of metallurgical companies. At present, it is required to mobilize companies' assets to solve the problems of their sustainable development, such as the problem of making development program of a metallurgical enterprise (company). The program is comprised of several various strategies of development: improvement of existing business processes (sales, procurement, production, maintenance of equipment, and others), production processes of various stages (coke, agglomerate, cast iron, steel, rolled products), implementation of problems of digital transformation, and others. Each strategy of the development program is comprised of projects specified by effect, amount of investments, changes expense items of operational budget related with operational costs of systems and processes, which should be improved by the project implementation, as well as indicator describing the risk of project implementation. One of the strategies of the development program can include multipurpose projects, implementation of which leads not only to changes in own performance indicators but also the changes in efficiency indicators of the projects of other (non-multipurpose) strategies of the development program. The case is considered when the management of development program includes management of total budget of the program and achievement of total target of the program (maximum effect of implementation of all projects of the development program). Herewith, the risk management and changes in operational budget are implemented at the level of management of project portfolio of individual program strategies (no restrictions for risks and changes in operational budget common for the development program). The stated formalization of the problems, the layouts of their decomposition and composition, and the developed procedures of solving separate subproblems are based on provisions and theory methods of system analysis and new section of discrete mathematics (network programming).

New method of multi-extremal optimization has been developed - a network programming method, which is based on the possibility to represent a complex function as a superposition of simpler functions. The paper deals with network (dichotomous) programming method and its application for solving multi-extremal and discrete optimization problems in Project Management. The main goal of the paper is to develop a new approach to solving knapsack problem and to solve a number of optimization problems of project building management. The optimal structure of the dichotomous representation is determined. On the basis dichotomous programming method new algorithm to solve a optimization problem is proposed

We analyzed program management tasks in the presence of soft dependencies between projects. Implementing soft dependencies reduces the time or cost of completing the immediate next project. The article formulates a problem and provides a solution, in which it is necessary to determine a set of executable soft dependencies under various conditions.

This paper considers the problem of motivating the reduction of project duration. The duration cuts of project works and the corresponding costs are given. A group incentive scheme is used to compensate for the costs. In this scheme, all works are partitioned into groups and a unified incentive scheme is applied for each group. Two types of unified incentive schemes are studied for groups, namely, linear and jump ones. The problem is to partition all project works into groups and choose an appropriate incentive scheme for each group by minimizing the total incentive fund. Solution algorithms are proposed based on determining the shortest path in the network. Special cases are also analyzed (partition with the minimum number of groups and partition with the maximum number of groups).

The improvement of management mechanisms for the formation and calendar planning of development programs is the most important direction for improving the productivity (achievement of goals) and efficiency (reduction of the amount of resources consumed) of the activities of metallurgical companies. Currently, it is necessary to ensure the mobilization of companies’ assets to solve the tasks of their sustainable development. The task of forming a program for the development of a metallurgical enterprise (company) is considered. The program includes several different areas of development: improvement of existing business processes (sales, supply, production, repair of equipment, etc.), production technologies of various stages (production of coke, agglomerate, cast iron, steel, rolled products), implementation of digital transformation tasks, etc. Each of the directions of the development program contains projects described by effect, size of investments, changes in the expenditure items of operating budget related to the costs of operating those systems and processes that the project is aimed at improving, as well as an indicator describing the risk of project implementation. One of the directions of the development program may include multi-purpose projects, the implementation of which leads not only to changes in its own performance indicators, but also to changes in the performance indicators of projects of other (non-multi-purpose) directions of such development program. The case is considered when management of the development program includes the management of the overall budget and achievement of overall goal of the program (the maximum effect from implementation of all projects). At the same time, project risk management and changes in the operating budget are implemented at the level of project portfolio management of individual program areas (there are no restrictions on risks and changes in the operating budget common to the development program). The stated formalizations of the problems, their decomposition and composition schemes, and the developed procedures for solving individual subtasks are based on the provisions and methods of the theories of system analysis and a new section of discrete mathematics (network programming).

The paper considers the problem of distribution of a limited resource for organizations. Namely, the mechanisms for the distribution of a limited resource in the case when the amount of the distributed resource depends on demand (the amount of applications). Three classes of mechanisms are being studied: mechanisms of absolute, direct and reverse priorities. In absolute priority mechanisms, the resource is allocated independently of the request. In direct priority mechanisms, the resource is distributed in proportion to the size of the request, and in reverse priority mechanisms, the resource is allocated inversely to the size of the request. An analysis has been carried out for various dependences of the amount of a resource on demand, and expressions for equilibrium situations have been obtained. The study shows that the introduction of the dependence of the value of the resource on the aggregate demand in some cases significantly complicates the analysis.

В статье рассматривается процедура формирования программ развития, включающих несколько направлений, каждое из которых содержит проекты, описываемые заданной эффективностью и размером инвестиций. The article discusses the procedure for the formation of development programs that include several areas, each of which contains projects described by a given efficiency and investment size.

A simple mathematical model examines the interrelation of three socio-economic formations - capitalism, socialism and communism. The authors introduce the factor of the public consciousness level. Its growth is a necessary condition for the transition to communism. However, the acceleration of the movement towards communism today requires a reduction in social wealth.

The paper considers the problem of complex activity coordinated planning, which is typical for hierarchical organizations. The authors have built a coordinated planning algorithm with integrated efficiency estimation for the case of separable costs. We also propose the method to account for the synergetic effects by analyzing the agent's activity occurrences in the vertex coverings of the complex activities' dependency graph.

The network programming method is based on the ability to represent the objective function and the system of constraints in the form of a superposition of simpler problems. It is convenient to represent such a superposition in the form of a network, the final vertices of which correspond to variables, and the remaining vertices correspond to simpler problems included in the superposition. The solution to the problem at the input vertex gives either an estimate (upper or lower) for the original problem, or an exact solution if the network representation structure is a tree.The article discusses the application of the method for an integer linear programming problem. To do this, the coefficients of the objective function are divided arbitrarily into m parts (according to the number of problem constraints). We get m tasks (each with one constraint). These tasks are called evaluative tasks. According to the main theorem of the network programming method, the sum of the values of the objective functions in the optimal solutions of estimation problems gives the upper estimate for the solution of maximum problems (or lower for the solution of minimum problems) for the original problem. The problem of choosing a partition of the coefficients of the objective function of the original problem so that the upper estimate is minimal is called the generalized dual problem (GDV). Note that the Lagrange minimization problem in the Lagrange multiplier method is a special case of the GDV. I.V. Burkova proved that GDV is a convex programming problem.The article proposes a method for solving the GDV based on the sequential improvement of the initial solution. At each step of the method, the system of linear inequalities is solved.

This paper considers the problems of adopting digital technologies in decision-making. Two types of digital decision technologies are described, namely, the direct (or traditional) technology with human decision-making based on computer-aided advisor systems and also the inverse technology with independent computer decision-making in which a human merely monitors the decision process. These technologies are compared with each other, and some applications are discussed.

The paper deals with an application of the network (dichotomous) programming method for solving multi-extremal problems and discrete optimization problems. The concept of a generalized dual problem is introduced and a theorem on its convexity is proved. Network programming method is used to build a business-forming project portfolio as well as an algorithm for solving completely dependent related projects is developed. Also, this method is used to solve the problem of building a business-supporting project portfolio for which the lower cost estimate is obtained. A computational experiment is carried out to evaluate the suggested algorithm, which showed that for large dimensions of the problem it is more effective than for solving the problem by linear programming methods.

The article discusses the mechanisms of smart management. These are management mechanisms that change people’s behavior in the direction necessary for society, foster honesty, responsibility, striving for development, etc. Smart mechanisms are being developed in our country - the theory of active systems, the theory of games with opposite interests, abroad - the theory of contracts, the theory of designing mechanisms. Four smart mechanisms considered: non-manipulability mechanisms - mechanisms under which it is beneficial to report reliable information (foster honesty); agreed mechanisms - mechanisms under which it is beneficial for performers to fulfill their obligations and plan (foster responsibility); advising mechanisms - develop recommendations (advice) to the decision maker. These can be software expert systems or a group of experts. There are two types - passive counselors and active counselors. A passive advisor provides the decision maker with information that is not responsible for its effectiveness. In the case of an active advisor, something like a competition arises between him and the operator, and depending on the effectiveness of the advice or the decision made by the decision maker, one of the parties is rewarded independently; developmental mechanisms - stimulate development (cost reduction, innovation). Examples of using smart mechanisms are given. The reasons for the slow implementation of innovations in management are discussed.

The paper proposes the development of the approach described in the ITIL-3,4 library to the formation by an IT provider of competitive IT assets (called “best practices”). The ITIL mechanism provides for regular benchmarking in order to determine the composition and values of indicators of critical factors for the success of the market area and the formation by the IT provider by methods of the theory of asset management, the functioning of which ensures the achievement/superiority of market values of these indicators. The difference between the proposed approach consists in replacing the “best practices” of management with procedures for solving optimization and game problems, the mathematical models and control mechanisms of which are based on mathematics, control theory, decision theory and game theory. The proposed approach makes it possible to overcome such disadvantages of “best practices” as the averaging of the conditions for their application and their insufficient efficiency. This is achieved due to the multivariate formulation of control problems, and the multi-structure and optimality of control mechanisms. In conclusion, it is indicated that it is legitimate to consider the developed models and management mechanisms as a separate new type of IT provider assets.

The network programming method is applied to the discrete problem of minimizing the costs of a project with a given duration. Two basic algorithms for solving this problem in the cases of independent and sequential jobs are described. More complex cases of project scheduling on networks (tree network and aggregated network) are treated by the sequential application of the basic algorithms. For the case of a build-up assembly network, a new method based on the following idea is proposed: an appropriate set of jobs is determined and their durations are fixated so that the resulting network becomes either a tree or an aggregated network. All possible alternatives for fixating the durations of jobs from such a set are considered and, for each alternative, the problem is completely solved. The best alternative among all possible ones is selected. The case of project scheduling on an arbitrary network is also studied.

Introduction. The construction of a high-quality comprehensive assessment of an object requires a more accurate and comprehensive accounting of both objective information about the object, subjective assessments, individual experience, intuition and knowledge of the head of the object. Aim. The aim of the work is to develop a modern approach to solving the problems of multi-criteria assessment and ranking based on the formation of a coherent structure of an integrated assessment. To do this, a choice is made of the structure of the complex assessment system (a dichotomous tree, the hanging vertices of which correspond to the estimated directions and the root vertex corresponds to the complex assessment) and the matrix convolutions in each (non-hanging) tree vertex are selected. In addition, requirements are developed for grading scales, a convolution tree, and convolution matrices, which allow one to clearly demonstrate the synergistic effect and give a description of the class of generalized median schemes implemented on the convex representation of binary trees. Materials and methods. The solution to these problems is based on the method of synthesis and generalization of existing approaches to building a comprehensive assessment and simulation when choosing the structure of a complex assessment system. At the same time, convolution matrices are determined as training options set by experts, such that for any training option, the score obtained by the system of assessment coincides with the expert score. Results. The conditions for the existence of a matrix in the form of inequalities for generalized estimates of training sub-options are obtained. These inequalities are associated with a graph whose vertices denote sub-options, and arcs reflect inequalities connecting generalized estimates. It is shown that if the graph has no contours, then the convolution matrix exists. An algorithm is proposed for determining generalized estimates of sub variants based on the obtained graph and the subsequent determination of the corresponding matrix. The substantiation of the requirements for assessment scales and the structure of the convolution tree is given. The requirements are obtained for the dimension and type of convolution matrices located in the nodes of the tree. The possibility of implementing a set of training data using some kind of integrated assessment mechanism is determined. Conclusion. The results obtained make it possible to formulate integrated assessment systems that provide flexibility of tuning to the preferences of decision makers, ease of calculation, and the ability to solve optimization problems of program formation on this basis.

Analysis of unsuccessful projects shows that the main reason is the violation of the time limits of the project. This implies the importance of project time management processes. It is known that measures aimed at reducing the time, as a rule, are associated with the occurrence of additional costs for performers. Naturally, the project manager must compensate for these costs, which occurs within the framework of the incentive system used. Given that engineering is a key area of the state economy, which largely determines its innovative development, it is very important to improve project management methods. It is shown how it is possible to carry out the development of a group incentive system aimed at ensuring a given value for reducing the duration of work. In this case, all the work, the duration of which must be reduced, is divided into groups and each such group has its own individual individual incentive system for all the works of this group. Naturally, such an incentive system is more expensive than an individual, but less than a universal one. The problem arises of designing group incentive systems that minimize the costs allocated to these goals.

Deployment of new service assets in operational IT environment is associated with the risk of disruption of the assets of its “basic” condition. Such disruptions cause failures of functioning services. To reduce the risk of disruption, the deploying service assets are divided by releases – sub-sets of service assets that will be embedded in the IT environment in one period. The traditional approach to formation and deployment of releases uses information on services structural properties to predict the number of failures due to the deployment of each service asset, each release and each application for deployment. As a result, the task of managing deployment of service assets is reduced to a sequential solution of three tasks: determining number of releases; determining composition of releases; building a release deployment schedule. The approach is based on the assumption that incorrect deployment of releases is manifested through the failure of services immediately after deployment. In practice, as a rule, this assumption is usually not fulfilled, because the use of various services by users who detect service failures is cyclical (daily, weekly, monthly, quarterly, annual). Many service failures can be detected by users in periods of time that are quite remote from the deployment time of the corresponding IT assets. The article contains the case where the managing process of IT provider configurations of the metallurgical company is well developed, i.e. its configuration database contains information on frequency of various services use at different periods of time. Information on dynamics of services application by users is used to predict time sequence of operating services failures due to deployment of each application. As a result, the task of forming and scheduling deployment of releases is formalized in the form of a discrete nonlinear programming task, the solution procedure of which allows simultaneous determination of number of releases, their composition and schedule for their deployment.