4. The generation of the area of feasible solutions by solving the tasks for optimal control of projects and production systems

By the implementation of management tasks as dynamic management tasks, where the solution is the function of time, it should be noted that the restrictions can also change in time. It happens as the characteristics of production system can alter in time, the changes can affect the schedule of supplies, the volume of resources allocated for the implementation of a certain project etc. The restrictions can be shown as follows:

$$\text{dim}\_1(t) $$

where M is the parameter or an expression with imposed restrictions, m1ð Þt and m2ð Þt is the restrictions set by the functions of time, m3ð Þt is the area of feasible values can also alter in time. 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 criteria and used by performing the operation of criteria compression.

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 calculating the value:

$$\sum\_{i=1}^{n} F\_i(m\_1(t), m\_2(t), m\_3(t), t) \tag{2}$$

where n is the number of restrictions.

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

4. All points xi ∈ ΛDi are studied for optimal factor, after that they are used to form an optimal set of solutions ΛP. The required sets are easily recovered from the labels of criteria in spaces.

5. The selection of the singular variant Xb , where X is the vector Xb ¼ ð Þ x1; x2;…; xN , from the Pareto-set is submitted to an expert, that has additional information that has not been

6. For an operational reaction to altering external factors we should perform several iterations for task solution (by modeling the deviations of forecasting values within the confi-

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

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

4. The generation of the area of feasible solutions by solving the tasks

By the implementation of management tasks as dynamic management tasks, where the solution is the function of time, it should be noted that the restrictions can also change in time. It happens as the characteristics of production system can alter in time, the changes can affect the schedule of supplies, the volume of resources allocated for the implementation of a certain

m1ð Þt <= ≤ M <= ≤ m2ð Þt , m1ð Þt <= ≤ M, M <= ≤ m2ð Þt , M ∈ m3ð Þt ,

where M is the parameter or an expression with imposed restrictions, m1ð Þt and m2ð Þt is the restrictions set by the functions of time, m3ð Þt is the area of feasible values can also alter in time.

for optimal control of projects and production systems

way xi ∈ Λ∗∗ make the set ΛDi

84 Digital Transformation in Smart Manufacturing

normal distribution law).

chart that is so widespread in management) [20].

project etc. The restrictions can be shown as follows:

.

formalized and neither taken into account in the model.

dence interval) and do that cyclically with the time period Δt.

If this value is equal to the sum of minimal or maximal values P<sup>n</sup> <sup>i</sup>¼<sup>1</sup> min=maxFiðm1ð Þ<sup>t</sup> ; 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 not as stiff and we can determine the feasible deviation of values (∓ Δ).

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 the following form:

$$J + \sum\_{i=1}^{n} K\_i F\_i(m\_1(t), m\_2(t), m\_3(t), t) \to opt\_\prime \tag{3}$$

where J is the criterial function.

#### 5. The problems of obtaining solutions as functions of time

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

Figure 4. The tree of management task solutions taking into account time factor X is the vector of variable values received in the solution of an optimization task, mΔt is the planning horizon, m is the number of task solutions, n is the number of solutions found on each step.

not just one solution, but a set of solutions that are Pareto optimal. So, the task solution will be a set of development paths that technically can be shown as a tree for each of the required parameter values (see Figure 4) that can be considered as Bayesian network.

The selection of a singular solution will be based on the choice of a path and on the potential of its implementation. The potential of each solution will be defined by chain rule [21]:

$$P\left(X^{(0)},\ldots,X^{(m)}\right) = \prod\_{j=1}^{m} P\left(X^{(j)}|X^{(j-1)},\ldots,X^{(1)}\right).\tag{4}$$

μð Þ¼ x<sup>2</sup>

σ2

in a new formula we add σ<sup>2</sup>

calculated by the formula [22]:

probable ones.

into the state sj, and pð Þ<sup>1</sup>

pð Þ<sup>1</sup> <sup>1</sup> ; <sup>p</sup>ð Þ<sup>1</sup>

<sup>2</sup> ;…; pð Þ<sup>1</sup> n � � <sup>¼</sup> <sup>p</sup>ð Þ<sup>0</sup>

programming (Bellman method) (see Figure 5).

Figure 5. Decision tree for PS path selection task or project implementation.

1 ð Þ σ<sup>1</sup> þ σ<sup>2</sup>

<sup>2</sup> <sup>¼</sup> D x½ �¼ M x<sup>2</sup> � � <sup>¼</sup> <sup>X</sup><sup>m</sup>

for x<sup>1</sup> (belonging to the interval σ in order to perform the validation for adequacy).

ment probabilities of a series of consecutive states s1, s2, ⋯, sn. If the probability pð Þ<sup>0</sup>

<sup>1</sup> ; <sup>p</sup>ð Þ<sup>0</sup>

where D x½ � is the dispersion, M x<sup>2</sup> � � is the mathematical expectation, <sup>x</sup>1<sup>j</sup> is the possible values

As a result, it is possible to define the probabilities of obtaining solutions and select the most

The use of the probability density functions for modeling deviation helps measure the achieve-

that we are placed in the state si and the state fully complies with the expected state (determined on the basis of previous stages), pij shows the probability of the transfer from the state si

> <sup>2</sup> ;…; pð Þ<sup>0</sup> n

and the management task adds up to the selection of a desired state from the set of possible states and the determination of a path (the set of delta states) to achieve this desired state. Therefore, it is possible to define the probabilities for obtaining decisions that will be taken into account for further selection of the most probabilistic ones based on the method of dynamic

� �

ffiffiffiffiffiffi <sup>2</sup><sup>π</sup> <sup>p</sup> <sup>e</sup>

> j¼1 x2 1j

<sup>i</sup> indicates the probability that the state si will be achieved. Then:

0

BBBB@

⋮ pn1

p11 p12 ⋯ p1n p12 p22 ⋯ p2n

> ⋱ ⋮ ⋯ pnn

1

CCCCA

⋮ pn2

� <sup>x</sup>�<sup>x</sup> ð Þ<sup>2</sup> <sup>2</sup> 2 σ2 1 <sup>þ</sup>σ<sup>2</sup>

<sup>2</sup> is the Gaussian perturbation of constant dispersion that is

<sup>2</sup> ð Þ (6)

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Time Factor in Operation Research Tasks for Smart Manufacturing

μð Þ x<sup>1</sup> (7)

<sup>1</sup> indicates

(8)

87

Therefore, by the planning horizon in mΔt and n solutions on each step we will obtain Q<sup>m</sup> <sup>j</sup>¼<sup>1</sup> <sup>n</sup> probabilities for leaf nodes in the built tree that should satisfy the following conditions P<sup>n</sup> <sup>i</sup>¼<sup>1</sup> P Xð Þ<sup>1</sup> i � � <sup>¼</sup> 1, <sup>P</sup><sup>n</sup> i¼1 P<sup>n</sup> <sup>j</sup>¼<sup>1</sup> P Xð Þ<sup>2</sup> ij � � <sup>¼</sup> 1, etc. for each solution step.

If we assume that all X are unique, then the implementation potential for each solution will be equal. However, in practice solutions can repeat. It is connected with the fact that we use the method of random search for solving a task; more than that, for modeling deviations we need a multiple solution of a considered task. In this case, the probability of a transfer from the state Xð Þ<sup>0</sup> into the state Xð Þ <sup>m</sup> will be determined by the sum of probabilities of repeated values, and this value will determine the probability of a transfer from one decision point to another one.

This probability will not be a random value since multiple calculations are performed, as parameters that are obtained based on forecast data can have random walk described by the functions of probability density; the latter ones are necessary to be used for generating new forecast values by multiple calculations.

$$\mu(\mathbf{x}\_1) = \frac{1}{\sigma\_1 \sqrt{2\pi}} e^{-\frac{\left(\mathbf{x} - \mathbf{x}\_1\right)^2}{2\sigma\_1^2}} \tag{5}$$

where σ<sup>1</sup> is the standard deviation, x<sup>1</sup> is the value obtained by forecasting. By a transfer to the consequent value the function will alter:

Time Factor in Operation Research Tasks for Smart Manufacturing http://dx.doi.org/10.5772/intechopen.73085 87

$$\mu(\mathbf{x}\_2) = \frac{1}{(\sigma\_1 + \sigma\_2)\sqrt{2\pi}} e^{-\frac{(\mathbf{x}-\mathbf{v}\_2)^2}{2\left(\sigma\_1^2 + \sigma\_2^2\right)}}\tag{6}$$

in a new formula we add σ<sup>2</sup> <sup>2</sup> is the Gaussian perturbation of constant dispersion that is calculated by the formula [22]:

$$
\sigma\_2^2 = D[\mathbf{x}] = M[\mathbf{x}^2] = \sum\_{j=1}^m \mathbf{x}\_{1j}^2 \mu(\mathbf{x}\_1) \tag{7}
$$

where D x½ � is the dispersion, M x<sup>2</sup> � � is the mathematical expectation, <sup>x</sup>1<sup>j</sup> is the possible values for x<sup>1</sup> (belonging to the interval σ in order to perform the validation for adequacy).

As a result, it is possible to define the probabilities of obtaining solutions and select the most probable ones.

not just one solution, but a set of solutions that are Pareto optimal. So, the task solution will be a set of development paths that technically can be shown as a tree for each of the required

Figure 4. The tree of management task solutions taking into account time factor X is the vector of variable values received in the solution of an optimization task, mΔt is the planning horizon, m is the number of task solutions, n is the number of

The selection of a singular solution will be based on the choice of a path and on the potential of

P Xð Þ<sup>j</sup> <sup>j</sup>Xð Þ <sup>j</sup>�<sup>1</sup> ;…; <sup>X</sup>ð Þ<sup>1</sup> � �

: (4)

<sup>1</sup> (5)

<sup>j</sup>¼<sup>1</sup> <sup>n</sup>

parameter values (see Figure 4) that can be considered as Bayesian network.

P Xð Þ<sup>0</sup> ; …; <sup>X</sup>ð Þ <sup>m</sup> � �

<sup>j</sup>¼<sup>1</sup> P Xð Þ<sup>2</sup> ij � �

P<sup>n</sup>

<sup>i</sup>¼<sup>1</sup> P Xð Þ<sup>1</sup> i � �

solutions found on each step.

86 Digital Transformation in Smart Manufacturing

<sup>¼</sup> 1, <sup>P</sup><sup>n</sup> i¼1 P<sup>n</sup>

forecast values by multiple calculations.

consequent value the function will alter:

its implementation. The potential of each solution will be defined by chain rule [21]:

<sup>¼</sup> <sup>Y</sup><sup>m</sup> j¼1

Therefore, by the planning horizon in mΔt and n solutions on each step we will obtain Q<sup>m</sup>

probabilities for leaf nodes in the built tree that should satisfy the following conditions

If we assume that all X are unique, then the implementation potential for each solution will be equal. However, in practice solutions can repeat. It is connected with the fact that we use the method of random search for solving a task; more than that, for modeling deviations we need a multiple solution of a considered task. In this case, the probability of a transfer from the state Xð Þ<sup>0</sup> into the state Xð Þ <sup>m</sup> will be determined by the sum of probabilities of repeated values, and this value will determine the probability of a transfer from one decision point to another one. This probability will not be a random value since multiple calculations are performed, as parameters that are obtained based on forecast data can have random walk described by the functions of probability density; the latter ones are necessary to be used for generating new

> 1 σ1 ffiffiffiffiffiffi <sup>2</sup><sup>π</sup> <sup>p</sup> <sup>e</sup>

where σ<sup>1</sup> is the standard deviation, x<sup>1</sup> is the value obtained by forecasting. By a transfer to the

� <sup>x</sup>�<sup>x</sup> ð Þ<sup>1</sup> <sup>2</sup> 2σ2

μð Þ¼ x<sup>1</sup>

¼ 1, etc. for each solution step.

The use of the probability density functions for modeling deviation helps measure the achievement probabilities of a series of consecutive states s1, s2, ⋯, sn. If the probability pð Þ<sup>0</sup> <sup>1</sup> indicates that we are placed in the state si and the state fully complies with the expected state (determined on the basis of previous stages), pij shows the probability of the transfer from the state si

into the state sj, and pð Þ<sup>1</sup> <sup>i</sup> indicates the probability that the state si will be achieved. Then:

$$\mathbf{P}\left(\mathbf{p}\_1^{(1)}, \mathbf{p}\_2^{(1)}, \dots, \mathbf{p}\_n^{(1)}\right) = \left(\mathbf{p}\_1^{(0)}, \mathbf{p}\_2^{(0)}, \dots, \mathbf{p}\_n^{(0)}\right) \begin{pmatrix} \mathbf{p}\_{11} & \mathbf{p}\_{12} & \cdots & \mathbf{p}\_{1n} \\ \mathbf{p}\_{12} & \mathbf{p}\_{22} & \cdots & \mathbf{p}\_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{p}\_{n1} & \mathbf{p}\_{n2} & \cdots & \mathbf{p}\_{nn} \end{pmatrix} \tag{8}$$

and the management task adds up to the selection of a desired state from the set of possible states and the determination of a path (the set of delta states) to achieve this desired state. Therefore, it is possible to define the probabilities for obtaining decisions that will be taken into account for further selection of the most probabilistic ones based on the method of dynamic programming (Bellman method) (see Figure 5).

Figure 5. Decision tree for PS path selection task or project implementation.

Each state is determined by a risk metric (a value that is calculated on the base of the probability pij depending on the path that we have taken to land at the examined state) and the dynamics of the change in the criterion value by the transfer from one special state into another one (see Figure 5).

By obtaining the solutions as the functions of time on each step of calculations the time step Δt becomes an important algorithm parameter. On the one hand, as a step we can choose the time between the decision points <sup>T</sup>ð Þ <sup>i</sup>þ<sup>1</sup> � <sup>T</sup>ð Þ<sup>i</sup> , from the other hand, by such approach the sensitivity of the system to altering external factors is decreasing (it becomes inertial). That is why, the selection of time step will be a trade-off between sensitivity and persistence of system. At the same time, time step can be an altering dimension (Δt ¼ f tð Þ) but it should be placed in the diapason <sup>τ</sup> <sup>≤</sup> <sup>Δ</sup><sup>t</sup> <sup>≤</sup> <sup>T</sup>ð Þ <sup>i</sup>þ<sup>1</sup> � <sup>T</sup>ð Þ<sup>i</sup> , where <sup>τ</sup> is the minimal time required for changing production capacity, reset of technological cycle etc. (system characteristic), <sup>T</sup>ð Þ <sup>i</sup>þ<sup>1</sup> � <sup>T</sup>ð Þ<sup>i</sup> is the time for the next decision point. There can be any number of solutions between decision points.

Underlying a new calculation is the output of values of a forecast parameter outside the bounder of the confidence interval �σ. On the other hand, works related to changing production capacity, production and procurement scheduling etc., bring additional expenses for enterprise (in general, we encounter the situations, when production capacity is to be increased first and decreased afterwards, that in some cases can be balanced, particularly, by stocks. Therefore, we should consider this task as a separate management task and use the algorithm shown in Figure 6.

The solution for the examined diapason <sup>T</sup>ð Þ <sup>i</sup>þ<sup>1</sup> � <sup>T</sup>ð Þ<sup>i</sup> by, for instance, joint consideration of the tasks of volume planning and procurement management will be production plan, the value of the given criterion (with a potential deviation diapason of decisions), the value of risk metrics and the volumes of changes in required parts and components taking into account possible

Figure 7. The solution results of volume planning and procurement management tasks based on the collected data about production system for a discrete production: (a) an example of production output volume for one of the products by the use of different forecasting methods, (b) the values of risk metrics (solid line) and progressive risk metrics (dotted line) connected with the use of planning data, (c) adjusted criterion value by the use of best forecast results and the corridor of possible deviations by the use of normal distribution for their modeling and its correlation with the retrospective databased criterion value, (d) the need in one type of parts taking into account possible deviations in production plan.

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89

The present chapter describes the approaches that thanks to the use of the concepts Industry 4.0 enable the formalization of the processes that are connected with the reasoning and

deviations from target production volumes (Figure 7).

6. Conclusion

Figure 6. The algorithm for defining the step Δt for time moment ~t, where J is the criterion value, k is the amount of work expenses for changing production cycle taking into account economic criteria.

Time Factor in Operation Research Tasks for Smart Manufacturing http://dx.doi.org/10.5772/intechopen.73085 89

Figure 7. The solution results of volume planning and procurement management tasks based on the collected data about production system for a discrete production: (a) an example of production output volume for one of the products by the use of different forecasting methods, (b) the values of risk metrics (solid line) and progressive risk metrics (dotted line) connected with the use of planning data, (c) adjusted criterion value by the use of best forecast results and the corridor of possible deviations by the use of normal distribution for their modeling and its correlation with the retrospective databased criterion value, (d) the need in one type of parts taking into account possible deviations in production plan.

The solution for the examined diapason <sup>T</sup>ð Þ <sup>i</sup>þ<sup>1</sup> � <sup>T</sup>ð Þ<sup>i</sup> by, for instance, joint consideration of the tasks of volume planning and procurement management will be production plan, the value of the given criterion (with a potential deviation diapason of decisions), the value of risk metrics and the volumes of changes in required parts and components taking into account possible deviations from target production volumes (Figure 7).

#### 6. Conclusion

Each state is determined by a risk metric (a value that is calculated on the base of the probability pij depending on the path that we have taken to land at the examined state) and the dynamics of the change in the criterion value by the transfer from one special state into another

By obtaining the solutions as the functions of time on each step of calculations the time step Δt becomes an important algorithm parameter. On the one hand, as a step we can choose the time between the decision points <sup>T</sup>ð Þ <sup>i</sup>þ<sup>1</sup> � <sup>T</sup>ð Þ<sup>i</sup> , from the other hand, by such approach the sensitivity of the system to altering external factors is decreasing (it becomes inertial). That is why, the selection of time step will be a trade-off between sensitivity and persistence of system. At the same time, time step can be an altering dimension (Δt ¼ f tð Þ) but it should be placed in the diapason <sup>τ</sup> <sup>≤</sup> <sup>Δ</sup><sup>t</sup> <sup>≤</sup> <sup>T</sup>ð Þ <sup>i</sup>þ<sup>1</sup> � <sup>T</sup>ð Þ<sup>i</sup> , where <sup>τ</sup> is the minimal time required for changing production capacity, reset of technological cycle etc. (system characteristic), <sup>T</sup>ð Þ <sup>i</sup>þ<sup>1</sup> � <sup>T</sup>ð Þ<sup>i</sup> is the time for the

Underlying a new calculation is the output of values of a forecast parameter outside the bounder of the confidence interval �σ. On the other hand, works related to changing production capacity, production and procurement scheduling etc., bring additional expenses for enterprise (in general, we encounter the situations, when production capacity is to be increased first and decreased afterwards, that in some cases can be balanced, particularly, by stocks. Therefore, we should consider this task as a separate management task and use the algorithm

Figure 6. The algorithm for defining the step Δt for time moment ~t, where J is the criterion value, k is the amount of work

expenses for changing production cycle taking into account economic criteria.

next decision point. There can be any number of solutions between decision points.

one (see Figure 5).

88 Digital Transformation in Smart Manufacturing

shown in Figure 6.

The present chapter describes the approaches that thanks to the use of the concepts Industry 4.0 enable the formalization of the processes that are connected with the reasoning and preparation of managerial decisions which are based on real statistical data that take into consideration the interaction of subsystems in production system. Therefore, together with the use of predictive models IIoT helps not only enhance the level of automation and reduce a certain part of personnel production expenses but also consider such factors as increasing power intensity and resources consumption of productions, inertness of integration and management processes in production systems, and the situations that are connected with repair actions, equipment mortality, procurement failures, change in demand and prices etc.

Author details

Leonid A. Mylnikov

References

824-836

Address all correspondence to: leonid.mylnikov@pstu.ru

Dec 2016;20(08):1640015-1-1640015-25

Choice: Pittsburgh, PA; 1996

2017;20(3):291-310

Wiley & Sons; 1966

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Perm National Research Polytechnic University, Perm, Russia

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We have investigated the question how to use and apply under existing conditions the approaches that search feasible and optimal solutions in the tasks of efficient management and planning (taking into account time factor). The changes that affect the setting and solution of tasks can be explained by the shift to automated and automatic enterprises, by the shift from mass production to single-part production. In this connection, the current situation requires operational rearrangement of ongoing production processes; we need to increase global economics mobility, i.e. the variability of external environment where production systems operate.

The approach that is described in the chapter is relevant as it tackles management tasks given as optimization tasks; besides, it helps deal with the phenomenon of NP is the completeness of obtained tasks.

The obtained results are sensitive to the quality of forecasts and lack time lags; more than that, we can observe a change in production volume that creates additional increased capacities for production system (related to the change in production schedule).

That is why, the shift to the concepts Industry 4.0 gives not only evident momentary advantages, but also outlines new areas for studies, i.e. the solution of tasks that take into consideration the inertness of production system and expenses that arise due to changes in production volume and risk metrics, that appear upon interaction with external systems (for example, delayed delivery, the delivery of faulty parts, return of goods etc.).

The development of mathematical formalization of these areas of studies can lead to additional effects in future and underlie the appearance of industrial concepts of next generations.
