1. Introduction

The decision-making process when managing innovatively active enterprises is a time-consuming task, which is complicated by the incompleteness of the initial information, the presence of many quality indicators (criteria) for evaluating the outcomes of alternative solutions to the problem, reducing the time for making decisions and increasing the requirements for the experience and qualifications of decision makers who, usually faces a number of difficulties [1]. First, these difficulties are associated with the high dimensionality of the tasks being solved: hundreds of parameters of the optimized model, tens, and sometimes hundreds, of performance criteria. Secondly, these difficulties are associated with the complexity of business models describing the activities of innovative enterprises: non-linearity, the difficulty of representing a system of constraints in the form of analytical relationships [2]. Even if in theory the problem being solved can be presented analytically, then, firstly, it turns out to be NP-complete, and therefore it cannot be guaranteed to be solved within a reasonable time, and secondly, the development of innovatively active enterprises requires constant restructuring and adaptation of production and logistics processes for the needs of a modern dynamically changing environment. Therefore, constant restructuring and adjustment of the model is required, which is long, laborious and increases the likelihood of errors [3-5]. In this paper, the principles of creating multi-agent systems based on fuzzy areas of preferences with the aim of optimizing management decisions in the dynamic conditions of the development of innovatively active enterprises are proposed [6,7]. The principles of the functioning of agents acting on the basis of the fuzzy choice function are developed, an algorithmic description of the behavior model of agents in a fuzzy information environment is given, a methodology for agent cooperation in order to find rational solutions is developed.

2. Materials and methods

Consider the following generalized formalization of agent classes:

• subjective agents — subjects of economic relations acting on the basis of their value system;
• objective agents — physical objects of the real world acting on the basis of formal rules;
• problem owner — a person who is interested in a comprehensive criterion for the effectiveness of the system and is responsible for the strategic decisions made;
• virtual agents are entities that do not have a direct embodiment of reflection in real economic processes, but allow finding optimal paths in the graphs of the corresponding production chains.

The agent preference function will be constructed by describing fuzzy preference areas. The following algorithm for solving the problem is implemented:

1) Define fuzzy scales for agent criteria.

2) For combinations of fuzzy gradations in the appropriate scales, assign fuzzy levels of preference.

3) Check that for all combinations of criteria values there is a level of preference, belonging to which is not lower than this.

4) If condition 3 is not fulfilled, then apply the refinement of the level of preferences using fuzzy analogues of quantitative methods for aggregating criteria, for example, a fuzzy weighted sum based on the principle of generalization.

To model the behavior of agents, the method of model events with a time-varying step was studied. To this end, the agent’s behavior is described in terms of the event planning relationship graph. In it, the vertices correspond to the planned events, and the arcs determine which events can cause the planning of other events. In case of ambiguity of planning, the function of fuzzy preferences of agents is used [8].

The event planning graph is unique within the objects of the corresponding classes inherited from the classes defined above. In addition to event planning graphs, the model contains graphs corresponding to the description of economic entities: production schedules, supply chains, and others. To exchange information between agents, a decentralized information architecture for agent interaction based on the bulletin board pattern has been developed. As a result, a complex of fuzzy methods, algorithms, simulation models, architectural solutions in the field of software allowed us to formulate a new methodology for finding rational management decisions for the development of innovatively active enterprises based on multi-agent fuzzy models.

During the research, a methodological approach is used, based on the use of effective ant algorithms for the dynamic optimization of processes in distributed non-stationary systems. Ant algorithms have been seriously studied by European scientists since the mid-90s of the last century. To date, good results have already been obtained for optimizing such complex combinatorial problems as the traveling salesman problem, the graph coloring task, the quadratic assignment problem, the optimization of network schedules, the scheduling problem, and many others [9-13].

One of the developed algorithms for optimizing the management of innovatively active enterprises is a “fuzzy” modification of the ant colony method, which can also be attributed to multi-agent algorithms. When solving this problem, this method is acceptable, because ants in this case are agents, since they meet the characteristics of agents:

1. Autonomy: agents, at least partially, are independent;
2. Limited representation: none of the agents has an idea about the whole system or the system is too complex for knowledge of it to have practical application for the agent;
3. Decentralization: there are no agents managing the entire system.

A number of researchers note that today ant algorithms are very competitive in comparison with other metaheurists and for some tasks give the best results.

In the future, it is planned to use the algorithms of transport logistics tasks. These tasks have many features and parameters. This paper discusses the resource management tasks of innovatively active enterprises in various interpretations. Most tasks of this type are solved using heuristic algorithms corresponding to the subject area. The developed algorithm based on the ant colony method with membership functions for fuzzy preference areas can solve such problems in a general form and can easily be modified for specific tasks of optimizing the management of innovatively active enterprises.

The whole algorithm of ant colonies is divided into two parts: in the first part, the function of preferences in fuzzy areas is determined — the number of pheromones, and in the second there are restrictions — a set of vertices where the agent can move from the one in which it is located.

Changing these two parts allows the ant colony algorithm to be applied to other tasks.

We have developed modifications of the ant colony method that allow us to search for rational solutions for various problems of managing inventory at innovatively active enterprises. In this paper, we developed a method for generating the objective function and restrictions based on fuzzy areas of preferences, and also describes the process of modifying the ant colony method to find rational solutions.

The proposed modifications of the ant colony method work on a variety of tasks, the parameters of which can be represented in the form of a graph. Each parameter of the described system is represented by a set of vertices containing specific values of this parameter. The parameters are ordered in random order and arcs are set in the graph that connect all the vertices of the values of one parameter with all the vertices of the values of the neighboring parameter. This structure allows us to present a solution to the problem of optimizing the management of innovatively active enterprises in the form of a search for a specific route in a graph (a specific alternative).

In the case of multi-criteria optimization, there is a preference function defined on fuzzy areas that displays fuzzy criteria values for each alternative. There is the problem of determining the amount of pheromone (the arc parameter for the ant colony method) recorded by the ant after passing the path to all the vertices of the decision graph. The solution to this problem is to enter a pheromone vector for each vertex. The dimension of this vector is determined by the number of criteria.

When an ant moves, only one “type of pheromone” can be used, i.e. at each vertex of the “decision graph” to operate with only one type of fuzzy weights from the set, then the ant will seek to find solutions that are optimal by this criterion. If, when choosing the next vertex, a convolution of the criterion is carried out (using the fuzzy weighted sum method, the combined method on fuzzy preference areas, etc.), then the ants moving along the generalized criterion obtained as a result of the convolution will look for an optimizing solution (minimizing in the case of the ant colony method) this criterion.

3. Results and Discussion

The main difficulty of the proposed method is the exponential increase in the number of runs of the algorithm with an increase in the number of criteria. To solve this problem, it is proposed to proceed not from the final solution, but from the parameters (in the «decision graph» the parameters are represented by many vertices). The idea of the method is the possibility of separating parameters according to criteria. For example, in the task of calculating the supply of resources to an innovatively active enterprise, it is possible to divide the parameters (specific names of resources) according to the criteria of efficiency and cost into expensive and high-quality resources that provide high reliability and cheap and lower-quality resources [14-20]. In this case, the variation of solutions according to a specific criterion can be provided by the corresponding group of parameters [20-23]. But it is not always possible to unambiguously determine the division of parameters into groups [24].

If we refuse to strictly divide the parameters into groups according to criteria, then for each parameter we can evaluate its effect on each criterion. As a result of a comprehensive study, an economic and mathematical multi-agent toolkit for multi-criteria optimization of the management of innovatively active enterprises was developed.

4. Acknowledgments

The reported study was funded by RFBR according to the research project № 18-00-00012 (18-00-00011) KOMFI