3. Solving management tasks with help of predictive models

Let us now consider a general task of formalizing management processes for project implementation in PS. This task can be handled as a task of defining decision points and a cyclic solution of prognostic models that are represented by optimized formalizations based on forecast data and elaboration done on each step of processing data in order to make consecutive iterations with new data, and calculation results.

In order for the tasks to be formalized as tasks of optimal control, we have to input a set of indices, variables and parameters of management [9], for instance, like: i is the supplier's index; j is the index of production system/stock (PS); m is the part index or the demand in materials; n is the index of end item; k is the index of production operation; g is the index of machine or instrument; p is the index of operation; t is the time; xijm is the number of parts m received from the supplier i for PS j; yjn is the number of parts n produced in PS j; rn is the number of returned items n for utilization; om is the number of reused parts or materials m; dm is the number of items or materials m sent to utilization; ref jm is the number of reused items or materials m in PS j; bdn is the binary variable that possesses the value equal to 1 in case if it can be repeatedly used for the item n and 0 if not; Δt is the time step; selln is the item's market price n; costjn is the item's production cost n in PS j; priceim is the price of the part m received from the supplier i; shipm=nij is the delivery cost of the part/ item m=n from the station i to the station j; invj is the storage cost in PS j; setdisn is the preparation cost to get the parts out of the item n; disam is the preparation cost to get the part m out for reuse; dispm is the utilization cost for the part m; refcostjm is the preparation/recovery cost of the part m for reuse in PS j; demð Þ<sup>j</sup> <sup>n</sup> is the need/demand in the item n, if there is the index j the consumer get then j; reqmn is the number of requested parts m required for the production of the item n; costeqpgj is the cost of the operation p on the equipment g in PS j; timeeqpgj is the time of operation performance p on the equipment g in PS j; partmpgj is the demand in parts/materials m in order to perform the operation p on the equipment g in PS j; eqpgj is the demand in the equipment g in order to perform the operation p in PS j; supmaxim is the maximum size of the batch of the parts m that can be delivered from the supplier i; supminim is the minimal size of the batch of the parts m that can be delivered from the supplier i; supmaxpartjm is the maximum potential number of parts and components m that can be delivered for production in PS j; supmaxeqjp is the maximum potential number of equipment units for the operation p in PS j; reusem is the maximum percentage of the parts m that can be reused.

The approach described above helps state a set of optimization tasks that can be considered both, as joint and separate tasks. Let us give the examples of feasible task formalizations:


The tasks can be subject to different restrictions:

In this case, each of tasks can be described by a separate criterion; the use of a reflexive approach enables their joint solution as a set of optimization tasks that have common param-

Figure 2. The interrelation of management levels and management tasks to be solved by using parameters and indicators

Let us now consider a general task of formalizing management processes for project implementation in PS. This task can be handled as a task of defining decision points and a cyclic solution of prognostic models that are represented by optimized formalizations based on forecast data and elaboration done on each step of processing data in order to make consecu-

In order for the tasks to be formalized as tasks of optimal control, we have to input a set of indices, variables and parameters of management [9], for instance, like: i is the supplier's index; j is the index of production system/stock (PS); m is the part index or the demand in materials; n is the index of end item; k is the index of production operation; g is the index of machine or instrument; p is the index of operation; t is the time; xijm is the number of parts m received from the supplier i for PS j; yjn is the number of parts n produced in PS j; rn is the number of returned items n for utilization; om is the number of reused parts or materials m; dm is the number of items or materials m sent to utilization; ref jm is the number of reused items or materials m in PS j; bdn is the binary variable that possesses the value equal to 1 in case if it can be repeatedly used for the item n and 0 if not; Δt is the time step; selln is the item's market price n; costjn is the item's production cost n in PS j; priceim is the price of the part m received from the supplier i; shipm=nij is the delivery cost of the part/ item m=n from the station i to the station j; invj is the storage cost in PS j; setdisn is the preparation cost to get the parts out of the item n; disam is the preparation cost to get the part m out for reuse; dispm is the utilization cost for the part m; refcostjm is the preparation/recovery cost of the part m for reuse in PS j; demð Þ<sup>j</sup> <sup>n</sup> is the need/demand in the item n, if there is the index j the consumer get then j; reqmn is the number of

3. Solving management tasks with help of predictive models

eters and use forecast-based data.

for developing decision support models.

80 Digital Transformation in Smart Manufacturing

tive iterations with new data, and calculation results.


• The restriction on the volume of reused parts, P <sup>j</sup> ref jmð Þt ≤ reusemð Þt omð Þt , ∀m, t, dmð Þt ≤ ð Þ 1 � reusemð Þt omð Þt , ∀m,t;

search in this case (for instance, due to time restrictions for decision making). If we use heuristic methods in task solutions, and heuristic elements complement combinatoric methods, it is getting more complicated to prove that the applied method is comprehensive.

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In practice hybrid algorithms are often used. Besides, the outcome of any algorithm work can be improved by building a joint solver. Due to the lack of specialized solution methods, for the obtained formalization we assume that we can use a developmental approach – the method of stochastic search. The drawback of developmental approaches is that in some cases the results and optimization time are dependent on the selection of initial approximation. This drawback can be eliminated by using as an initial approximation the solution that was worked out by experts. That is why, as a universal solution we suggest to use the method of stochastic search taking into consideration expert knowledge and indistinct preferences. However, in this case we need to direct attention to the fact that for some tasks we can obtain formalizations that already have methods of their solution. Hence, the decision about what method to apply should be taken dependent on the targets, i.e. how accurate the solution is expected to be and whether we have time restrictions for solution search (the methods of stochastic search can be limited in time required for solution search, which is crucial in integrated systems and IIoT

In heuristic methods of random search we can distinguish two big groups: the methods of random search with learning and developmental programming [18]. In practical use the methods differ in convergence speed and the number of iterations required for search of a feasible solution (several methods, for example, genetic methods, ensure finding an extremal value, but not obligatorily an optimal one). The complexity of selection task is that the efficiency performance of certain methods of stochastic search (in particular, genetic algorithm) is determined by their parameters. As an example let us examine the application of the method of random search with inhibits (Pareto simulated annealing) [19]; along that, we take into account the set values, that were obtained by forecasting during the modification of task for work with restrictions. Before we start perform numerical calculation we need to determine the area for feasible solutions. The algorithm will consist in five steps and an additional sixth step; the latter step allows solve tasks with the restrictions set by functions and forecast values with

the set accuracy and the criterion that can also use the values obtained by forecasting.

quantities. Consequently, the algorithm has the following sequence of steps:

algorithm). Depending on the certain task, the value N∗∗ can alter.

2. Find N∗∗ points for each parameter xi ∈ Λ∗∗, scattered in the spaces Bð Þ <sup>N</sup>

the use of expert knowledge, and use these points as an initial approximation.

Let us now consider the search option of parameter values xi, i ¼ 1, N as points in space Bi Let us assign Λ∗∗ to the set of all points xi, that comply with the task restrictions:

capacity of the finite set Bð Þ <sup>N</sup> , N is the number of components in the vector of unknown

1. Set N∗∗ is the requested number of points from the set Λ∗∗ (N∗∗ is the parameter of

<sup>i</sup> ; <sup>j</sup> <sup>¼</sup> <sup>1</sup>, N∗∗ n o (that are included in the area of feasible values), where <sup>N</sup>∗∗ is the

<sup>i</sup> randomly or by

The methods of heuristic search are, in general, incomplete.

that operate in real time).

<sup>Λ</sup>∗∗ <sup>¼</sup> <sup>x</sup>

ð Þj <sup>i</sup> <sup>∈</sup>Bð Þ<sup>j</sup>

• etc.

The obtained tasks in their general form refer to a class of multi-parameter tasks with nonlinear restrictions. In such tasks a part of parameters is set by time functions. The outcome of the solution of such tasks will be the function of time as well (by numerical solution in a table form). Since today we lack analytical methods to solve such tasks, we will build then the solution of this task on multiple cyclic determination of numerical solutions of a multiparameter optimization task with the time period Δt ≤ min <sup>i</sup>¼<sup>1</sup>, <sup>n</sup> <sup>T</sup>ð Þ <sup>i</sup>þ<sup>1</sup> � <sup>T</sup>ð Þ<sup>i</sup> that determines the accuracy of the description of the required function (see Figure 3).

Taking gradient calculation for finding solution was one of the first approaches to develop solution methods (gradient search method with the split of the step метод градиентного поиска с дроблением шага, steepest descent method, conjugate direction method, the Fletcher-Reeves method, the Davidon-Fletcher-Powell method). By these the goal function has to be differentiated two times and convex. The Newton's method and his modification the Newton–Raphson method is widespread. These methods also set the requirements to the goal function to be differentiated two times and be convex. Besides, these methods are sensitive to the selection of initial value. Moreover, in obtained optimization tasks we the cases can appear that are related with multiextremality, non-convex restrictions, multicoupling of the area of feasible solutions etc., and these methods cannot handle that appropriately. Modern methods can in general be split into three groups [17]: cluster methods, the methods of restrictions' distribution, metaheuristic methods. By choosing the solution method it is important to consider that the most significant feature of combinatoric optimization methods is their completeness and comprehension. A complete method ensures the finding of the task solution if it exists. However, the application of these methods can bring difficulties by a big dimension of search space, and we might not have sufficient amount of time that will be required for

Figure 3. The scheme that clarifies the principle of defining calculation points (special states) by implementing projects in PS.

search in this case (for instance, due to time restrictions for decision making). If we use heuristic methods in task solutions, and heuristic elements complement combinatoric methods, it is getting more complicated to prove that the applied method is comprehensive. The methods of heuristic search are, in general, incomplete.

• The restriction on the volume of reused parts, P

parameter optimization task with the time period Δt ≤ min

accuracy of the description of the required function (see Figure 3).

The obtained tasks in their general form refer to a class of multi-parameter tasks with nonlinear restrictions. In such tasks a part of parameters is set by time functions. The outcome of the solution of such tasks will be the function of time as well (by numerical solution in a table form). Since today we lack analytical methods to solve such tasks, we will build then the solution of this task on multiple cyclic determination of numerical solutions of a multi-

Taking gradient calculation for finding solution was one of the first approaches to develop solution methods (gradient search method with the split of the step метод градиентного поиска с дроблением шага, steepest descent method, conjugate direction method, the Fletcher-Reeves method, the Davidon-Fletcher-Powell method). By these the goal function has to be differentiated two times and convex. The Newton's method and his modification the Newton–Raphson method is widespread. These methods also set the requirements to the goal function to be differentiated two times and be convex. Besides, these methods are sensitive to the selection of initial value. Moreover, in obtained optimization tasks we the cases can appear that are related with multiextremality, non-convex restrictions, multicoupling of the area of feasible solutions etc., and these methods cannot handle that appropriately. Modern methods can in general be split into three groups [17]: cluster methods, the methods of restrictions' distribution, metaheuristic methods. By choosing the solution method it is important to consider that the most significant feature of combinatoric optimization methods is their completeness and comprehension. A complete method ensures the finding of the task solution if it exists. However, the application of these methods can bring difficulties by a big dimension of search space, and we might not have sufficient amount of time that will be required for

Figure 3. The scheme that clarifies the principle of defining calculation points (special states) by implementing projects in PS.

<sup>i</sup>¼<sup>1</sup>, <sup>n</sup>

dmð Þt ≤ ð Þ 1 � reusemð Þt omð Þt , ∀m,t;

82 Digital Transformation in Smart Manufacturing

• etc.

<sup>j</sup> ref jmð Þt ≤ reusemð Þt omð Þt , ∀m, t,

<sup>T</sup>ð Þ <sup>i</sup>þ<sup>1</sup> � <sup>T</sup>ð Þ<sup>i</sup> that determines the

In practice hybrid algorithms are often used. Besides, the outcome of any algorithm work can be improved by building a joint solver. Due to the lack of specialized solution methods, for the obtained formalization we assume that we can use a developmental approach – the method of stochastic search. The drawback of developmental approaches is that in some cases the results and optimization time are dependent on the selection of initial approximation. This drawback can be eliminated by using as an initial approximation the solution that was worked out by experts. That is why, as a universal solution we suggest to use the method of stochastic search taking into consideration expert knowledge and indistinct preferences. However, in this case we need to direct attention to the fact that for some tasks we can obtain formalizations that already have methods of their solution. Hence, the decision about what method to apply should be taken dependent on the targets, i.e. how accurate the solution is expected to be and whether we have time restrictions for solution search (the methods of stochastic search can be limited in time required for solution search, which is crucial in integrated systems and IIoT that operate in real time).

In heuristic methods of random search we can distinguish two big groups: the methods of random search with learning and developmental programming [18]. In practical use the methods differ in convergence speed and the number of iterations required for search of a feasible solution (several methods, for example, genetic methods, ensure finding an extremal value, but not obligatorily an optimal one). The complexity of selection task is that the efficiency performance of certain methods of stochastic search (in particular, genetic algorithm) is determined by their parameters. As an example let us examine the application of the method of random search with inhibits (Pareto simulated annealing) [19]; along that, we take into account the set values, that were obtained by forecasting during the modification of task for work with restrictions. Before we start perform numerical calculation we need to determine the area for feasible solutions. The algorithm will consist in five steps and an additional sixth step; the latter step allows solve tasks with the restrictions set by functions and forecast values with the set accuracy and the criterion that can also use the values obtained by forecasting.

Let us now consider the search option of parameter values xi, i ¼ 1, N as points in space Bi Let us assign Λ∗∗ to the set of all points xi, that comply with the task restrictions: <sup>Λ</sup>∗∗ <sup>¼</sup> <sup>x</sup> ð Þj <sup>i</sup> <sup>∈</sup>Bð Þ<sup>j</sup> <sup>i</sup> ; <sup>j</sup> <sup>¼</sup> <sup>1</sup>, N∗∗ n o (that are included in the area of feasible values), where <sup>N</sup>∗∗ is the capacity of the finite set Bð Þ <sup>N</sup> , N is the number of components in the vector of unknown quantities. Consequently, the algorithm has the following sequence of steps:


3. For finding the solutions xi ∈ ΛDi (ΛDi is the set of feasible points) apply one of the heuristic methods of stochastic search. For this purpose, the point xi ∈ Λ∗∗ is taken as a base point, and based on this point we build new points belonging to Λ∗∗ where the criterion values are better than in a base point. Even if one such point is found, its base is used then for finding new values etc., and next search is done. All the points found this way xi ∈ Λ∗∗ make the set ΛDi .

The use of several criteria and a big number of restrictions often leads to the situation that we obtain an empty area of feasible values or the solution shows some deviations. In any case, in PS management tasks the final decision is taken by the expert. That is why, the restrictions can be presented by the functions F mð Þ <sup>1</sup>ð Þt ; m2ð Þt ; t that can be represented in a form of additional

In case of discrete set values or if restrictions are set as an area of feasible values, the function F mð Þ <sup>1</sup>ð Þt ; m2ð Þt ; t or F mð Þ <sup>3</sup>ð Þt ; t becomes piecewise-set. Hence, the values that belong to a feasible interval are maximum high by considering a maximization task, and the others become maximum low and vice versa by considering a minimization task. In general, for the consideration of all types of restrictions in one record the function can be written as F mð <sup>1</sup>ð Þt ; m2ð Þt ; m3ð Þt ; tÞ: The membership with the area of feasible values can be validated then by

m2ð Þt ; m3ð Þt ; tÞ dependent on the type of the considered task (minimization or maximization), then it will belong to the area of feasible values. In practice, the restrictions can be considered

Such approach helps add restrictions to a criterial function as additive components that allows get rid of restrictions and apply for solution the methods that do not work with restrictions. Since restrictions can be destroyed in this case, so the obtained functions are to be ranged with help of weight coefficients K. As a result, we receive a final setting of the task for extremum in

By solving tasks of optimal control taking into account time factor and some discrete time step Δt the solution will be a set function presented in a table form. In this case, the system interacts with the external environment and the found solution can be not achievable due to the changes of external or internal factors. According to Bayes' theorem [21] the probability of a successful transfer to another state (to a new solution) will depend on the previous state (the state that we are placed now). Hence, for selecting the path for project development it is useful to consider

Fið Þ m1ð Þt ; m2ð Þt ; m3ð Þt ; t (2)

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KiFiðm1ð Þt ; m2ð Þt ; m3ð Þt ; tÞ ! opt, (3)

<sup>i</sup>¼<sup>1</sup> min=maxFiðm1ð Þ<sup>t</sup> ;

criteria and used by performing the operation of criteria compression.

Xn i¼1

If this value is equal to the sum of minimal or maximal values P<sup>n</sup>

not as stiff and we can determine the feasible deviation of values (∓ Δ).

5. The problems of obtaining solutions as functions of time

<sup>J</sup> <sup>þ</sup>X<sup>n</sup> i¼1

calculating the value:

the following form:

where J is the criterial function.

where n is the number of restrictions.


As a result, we receive altering in time span (corridor) of potential solutions for each time period. At the same time, as several functions describe the parameters that are set by forecasts, where accuracy depends on the planning horizon, we can encounter the case, when the obtained values can fluctuate either towards the increase or the decrease. Such behavior will bring additional organization expenses for PS; however, it is possible to manage such behavior (smoothly adjust the altered values) by changing the dimension within the obtained corridors and the time step Δt (as a rule, such deviation is described by a stochastic variable that obeys normal distribution law).

In the result of the solution we can determine the diapasons and the values of the values that can be presented in a suitable way to the decision-maker (for instance, in a form of the Gantt chart that is so widespread in management) [20].
