1. Introduction
The development of modern production and economics is generally based on newfound knowledge and scientific achievements that are integrated into technologies and products.
At the same time, companies are very responsive to any fluctuations in the market and consumer preferences. The situation on the market changes very rapidly, and the number of competitive products is vast. The companies have to launch new developments to catch up with modern trends, satisfy consumer preferences, and create new market segments. They should also bear in mind that product life time reduces as a result of increasing modifications and improvements in production and economic systems (PES). So, management sector that deals with innovation projects, especially in smallscale companies, should always make decisions promptly and ensure high quality in their products. In real project and PES management, we encounter an increasing number of statistical data, lack of universal algorithms, and the sufficient software to operate them, and besides, there are problems for which the current solutions are insufficient. It is difficult to describe the relationship among projects, PES, and the environment; therefore, live data are essential for effective management in the decisionmaking process.
The problem of management decisionmaking in PES projects was initially taken up by AlbertKalmes, as a problem of inventory and statistics in factories and commodity production; later on, Frederick Taylor and Henry Gantt laid the foundations for the methods of planning [1]. Mathematical calculations in this area are connected with system analysis (introduced by J. Von Neiman and L.V. Kantorovich [2]). Economical aspects are considered in this approach in terms of pricing, production planning, sequence scheduling, fixed price and timeandmaterials cost, and procurement management. These aspects are dependent on market segmentation and internal structure of PES, well known as WagnerWhitin algorithm that relates these aspects with market. The idea to integrate market selection and production planning was introduced in Ref. [3] and up till now, this problem has not been solved. It is NPcomplete problem that can be solved only if we fix separate factors (in particular, Jean Tirole has successfully solved management task for markets segmentation).
In 1995, Pepall introduced Game theory [4] to describe duplicates and innovations; this theory considers the relationship among projects. Such an approach triggered change management. Today, this idea is used in innovation management in terms of agent simulation and forecasting.
In the 1950s, mathematical and algorithmic calculations helped experts work out methods of planning and management known as Justintime; this method still enjoys high popularity. Main achievements in the late 1960s are connected with the works of Oliver White, who suggested that production, supply, and sale departments can be considered altogether in automated industrial enterprises. In his publications and in the periodicals of American Production and Inventory Control Society (APICS), we can find the algorithms of planning, which are known today as MRP. In the 1970s, Eli Goldratt in Israel worked out the method OPT. The modification of the algorithms of planning MRP~II was considered as the final achievement of all these methods till the beginning of the 1980s. The idea of computerbased integrated production CIM appeared in the first part of the 1980s, due to the integration of flexible manufacturing and efficient management. The US Department of Defence introduced, in the 1980s, CALS methods to ensure that all operations with orders, production development and organization, supplies, and operation of military technique were efficiently managed and planned. In beginning of the 20th century, the ideas of intellectual enterprise were introduced [5]; at that time, multiagent systems that were developed to consider such factors as autonomy, external factors’ dependency, flexibility, proactivity, social factor, and efficient intellectual management factor were not studied thoroughly enough to use them in complex information systems. In such an approach, it is difficult to ensure effective interaction among the parts of PES at an industrial enterprise. In particular, D.A. Novikov contributed greatly to the development of this area with his theory of active systems. This issue was also addressed by R.K. Sah and J. Stiglitz who proved the necessity of building complex structures in collective decisionmaking, for instance in organizational hierarchies.
2. Mathematical formalization of innovation in production and economic systems
The development of conceptual bases in management and simulation plays a significant role in PES and project management as it converts knowledge from object perspective into action [6].
Yu.A. Zelenkov introduced tuple description of goals and current possibilities of PES projects:
where
Within project approach in PES, it is recommended to use general purpose tuple [7]:
where
Sufficient management [8] requires formalized description of tuple parts (resulting in a lower degree of ambiguity).
In management, we should take into account that project and system develop over time and affect multiple PES. Therefore, it makes sense to consider the models for different points of time, levels, management types, and project stages that lead to necessity to study project identification and define decision points.
Such a task can be illustrated by determining managerial vector parameters
By the set of parameters, decision points can be defined by PES data (equipment service intervals and internal technology cycles, etc.), statistical data, or forecasting data that describe a project or projects (the parameters of sales volume and price change, etc., are presented by innovation curves), see Figure 2.
As a result, each decision point will be given a model that altogether will form a tuple
This way to form a tuple,
Hence, project time management can be reduced to tuples formalization
Forecasting tasks and the description of time series are studied by many authors, and there are many methods to solve this problem (mathematical regression model, functional description of parameters by innovation, and Scurves).
For mathematical formalization, we can refer to the scheme illustrated in the Figure 4.
Formalization can be presented by selecting most efficient way to developing market segmentation when same project (a development way) can enter different markets (particularly, B2N, and B2C). This task has the following mathematical formula [10]:
where
The selection of PES where this project will be implemented is another example (project can be transferred for implementation to existing PES or can be implemented independently by creating new legal entities). Based on PES tasks, we can deal with the task of return maximization from project implementation or handle the task of reducing production time. Therefore, we obtain two models.
First model is built for mitigating the production cost:
where
Second model is designed for mitigating time required. We need to note that certain operations can be performed simultaneously (see Figure 5).
The model for mitigating the time required can be given as follows:
where
In PES, project management is aimed at return optimization [11] via portfolio selection for goods. At the same time, not all the economically justified goods can be produced at each technological enterprise.
In order to cover these particular features, we need to give sound suggestions based on the set of criteria. For instance, criteria function and limitations will be given for volume scheduling of production planning as follows:
(7) 
where
Otherwise, criteria function can be written the following way:
by
The amount of costs
If it is necessary to consider these costs and the time required Eq. (6), we can complement the model with respective criteria of type Eq. (5).
The costs are specified by technological charts of product
Operation number  Operation name  Operation cost  Previous operations  Subsequent operations  Performance time  Need in parts  Need in materials  Need in equipment 


Forecast data are used for market conditions. Therefore, value scheduling parameters are
As a result, we receive a portfolio and product release schedule that stipulate the amount of expected return from one product and accumulated effect from a released group of products.
This way of formalization helps distinguish projects in terms of specific features of PES and expected change of market conditions or other critical project parameters that were used in criteria function.
However, we must emphasize that on each stage critical for the task to be tackled (time sequence is restricted by decision points), we might need multiple solution of the problem as a number of parameters is determined by forecast data, and the situation can be changed over the time.
Therefore, the problem can be solved by making a table with time function. Due to lack of analytical methods that can be used nowadays to solve obtained tasks, we suggest use multiple cyclic numerical solution with time period
Received solutions and time sequence selection can require additional research though, as we can encounter periodical change of production volume that leads to additional expenses for preparation and/or modification of production system.
Despite all the advantages of mathematical programming, in general, it is not easily solved (especially in case of multiple criteria). Such tasks are considered as NPcomplete problems (for instance, for market segmentation task [3]). Due to forecast errors, complex tasks obtained by mathematical programming can be solved by approximate methods. That is why it is very important to study sensitivity of gained solutions to the level of market and PES parameters’ deviation and take into account production and planning risk evaluation; the stipulated parameters can have Markov property (Markov process) and can be designed by MonteCarlo method.
Due to considerable restrictions nowadays, we can take advantage of other ways to formalize such groups of tasks. A vast amount of Nobel laureates focus on this problem (L.V. Kantorovich, 1975; R. Solow, 1987; H. Markowitz, 1994; J. Stiglitz, 2001; J. Tirole, 2014). Besides, management and sufficient formalization principles in management and applications greatly contribute to existing approaches and theories.
3. Risk evaluation
The analysis of gained results plays a significant role in managerial decisionmaking. Many authors make big efforts to tackle tasks with risk analysis of segregate solutions. For example, for project portfolio risk evaluation, we can use capital asset pricing model (CAMP) introduced by Sharpe [13], Lintner [14], and Mossin [15] based on the theory of Markowitz described in Refs. [16, 17]. For risk evaluation, we can also use the approach covered in Ref. [18], when we use functionbased parameters obtained by forecast margins.
Overtime consideration of parameters makes it possible to mitigate risks associated with the selection of innovation projects (managerial and organizational), for which membership function may be identified for every moment of time.
In this case, the expected return of product portfolio can be determined as is evident from:
where
Correlation factor can be calculated by the formula:
Then the risk evaluation for
Forecast data can be calculated by the formula [19]:
and
where
For retrospective data:
where
Correlation ration can be defined upon statistical data (Table 2) and Slope One algorithm [20].
Period (day, month, quarter, year, …)  Product 1  Product 2   Product   Product 

1  
      
In this case, the angle cosine between vectors formed by columns of values for each product (itemtoitem algorithm) can be used as the correlation factor:
where
These calculations can be subsequently used to fill in the product correspondence table.
In this case, portfolio will be chosen on the assumption that the expected return can be also determined by the following formula [21]:
where
4. Forecasting risks
When we use forecasts in decision making, we face risks on whether the forecasts are reliable and how the result will guarantee the quality of decisionmaking. Forecasting can be performed only by a certain degree of assurance; however, longterm forecasts produce low degree of accuracy (the intervals of potential deviations will increase). The magnitude of deviations can be calculated for normal distribution parameters based on the maximum margin of error when forecast can be regarded as accurate [18]:
where
Let us find
The fuzzy set
Let us assume that a risk is calculated for a certain set of project parameters taking into account all risk factors, i.e., multivariable risk:
where let us say,
Furthermore, let us suppose that risk is measured over a certain risk set:
where
Defined index that is used in this formula should be determined by either expert evaluations or forecast margins generated by diverse methods; all these margins are based on various original data. Hence, we use these data to define margin range of an interested parameter (i.e., risk measured by this method will uniquely be placed in the range, that generates fuzzy set), and membership function is built on Gaussian function (used by the description of normal distribution law).
To define function parameters, let use Gaussian function
where
For the parameters, that do not obey normal distribution law, we can use triangular functions, generic Bell function, and sigmoid function for asymmetric distributions, etc.
Based on membership function and taking into account potential risky events that influence each of the margins, we measure risk evaluation margin [18] for the described parameter value:
where
We should take into consideration that obtained risk evaluations do not consider the risks of previous stages.
Such forecasting risk evaluation can be applied only in the case if we know all the parameter values that we need to assess; that is a disadvantage of this method.
5. Conclusion
A simulated model can help forecast features and behavior of object of inquiry both inside the area, where the model is simulated, and (by proved application) outside this area (forecasting role of a model); manage the object by selecting most efficient modelbased impacts (managerial role); recognize the phenomenon or the object that was used for simulating the model (cognitive role of a model); obtain skills to manage the object by using the model as a training simulator or a game (training role); and enhance the object by modifying and testing the model (project role).
In practice, the stipulated task management in PES helps design simulation models for certain tasks avoiding NPcomplete problem (for instance, WagnerWhitin algorithm); furthermore, the use of sequential stagetostage transitions of forecast parameters or production cycles as described in Ref. [7] as crucial points in decisionmaking helps to avoid infinitehorizon problems [22] and exclude innovative regression in PES introduced by the corresponding member of RAS D.A. Novikov [23].
The described approach for management decisionmaking helps study PES processes at any accuracy degree. At the same time, the model complies with each management algorithm or system behavior and assesses risk margin for decisionmaking models.