2. Methodology aspect of management task setting in production systems

Production system is regarded as management object that is placed in a state space. The coordinates on this n is the dimensional space are represented by the management parameters that are considered significant for achieving the targets, and their values describe the current state and remoteness from the selected targets.

If we mark target goal indexes by the vector Pp, and the current state by the vector Pa, we will receive a mathematically measurable metric Pp; Pa that shows how the current position deviates from the goal position that is deemed a sign of progress for project implementation (the end of implementation, Pp ¼ Pa). However, to know the metrics Pp; Pa is not enough for management, we also need to know the vector of the parameters Y that greatly affect the state of project and consist of the values that describe project, production system and the environments in which project is implemented as well as dynamics of change and prognostic values of all these parameters. It should be noted, that the achievement of the goal values Pp ¼ Pa does not always mean the achievement of the vector values Y expected for this state.

In management tasks values and parameters can be classified in four groups [10]: parameters and values that describe a current state Pð Þ<sup>i</sup> <sup>p</sup> , values and parameters that describe the action (external factors and control action – Y ¼ A ∪ Θ, the A is the set of control actions, Θ is the set of environment values), values and parameters that describe a goal state Pð Þ<sup>i</sup> <sup>a</sup> , values and parameters that describe the output of system operation by shifting from the state Pð Þ<sup>i</sup> <sup>p</sup> into Pð Þ<sup>i</sup> <sup>a</sup> - R and time Tð Þ<sup>0</sup> .

Therefore, management has to use an automaton where the consecutive state is defined by experts based on the current state and the state that was planned to be achieved on the previous stage and the time when it has to be done – Pð Þ<sup>0</sup> <sup>p</sup> ; Pð Þ<sup>0</sup> <sup>a</sup> ; <sup>T</sup>ð Þ<sup>0</sup> , Pð Þ<sup>1</sup> <sup>p</sup> ; Pð Þ<sup>1</sup> <sup>a</sup> ; <sup>T</sup>ð Þ<sup>1</sup> , …, Pð Þ <sup>n</sup> <sup>p</sup> ; Pð Þ <sup>n</sup> <sup>a</sup> ; <sup>T</sup>ð Þ <sup>n</sup> . In order for a new state to come, action <sup>A</sup>ð Þ<sup>i</sup> has to be defined. We can determine such action with help of the production system model that implements innovation projects w<sup>j</sup> ¼ f g U; S , where U is the vector of management parameter, S is the set of project resource needs, j is project number.

This approach helps work out hierarchically coordinated managerial decisions by taking into consideration system-interrelated external and internal factors that interact. Management process is considered then as a holistic undetermined process.

In general, the model can be presented in a form of a tuple:

$$\psi = \left\{ Y, P\_p, P\_a, T, R, \varphi \right\} \tag{1}$$

Decision points can be defined in case if we know the set of controlled parameters [12], and have additional information that characterizes the production system that we manage (equip-

The setting of management tasks taking into account time factor Tð Þ<sup>i</sup> leads to formalizing the models <sup>Y</sup>ð Þ<sup>i</sup> ! <sup>M</sup>ð Þ<sup>i</sup> <sup>A</sup>; <sup>Θ</sup>; <sup>T</sup>ð Þ <sup>i</sup>�<sup>1</sup> , Tð Þ<sup>i</sup> � �. The structure of the model sets formal interrelations between its parameters, and on each step the type of the model will depend on the managerial task that we consider (whether we forecast properties and behavior of the investigated management project; or when dealing with object management we select best actions by testing them on

The model itself can use both non-causal (component-oriented) and causal (block-oriented) modeling, and model components can set requirements to their development tool (for example, the possibility to 1) work with big data volumes set by time series 2) use the methods that are applied for incomplete data 3) solve tasks set in a form of mathematical programming 4)

The specialization of models Yð Þ<sup>i</sup> brings the problem of choosing approaches and ways for formalization based on the set of already known approaches, ways, methods and models [14]

For the implementation of each project in the considered production system, the model formation that is presented in a general form is as follows f g <sup>R</sup>; <sup>w</sup>; <sup>A</sup>; <sup>Θ</sup> ! <sup>M</sup>ð Þ<sup>i</sup> <sup>A</sup>; <sup>Θ</sup>; <sup>T</sup>ð Þ <sup>i</sup>�<sup>1</sup> , Tð Þ<sup>i</sup> � � !

n o, where <sup>P</sup> is the vector of external parameters that exert impact on the system,

Despite the apparent simplicity of the approach, underlying this approach is a necessity to work out managerial decisions taking into account different levels (institutional, managerial, technical) and management types (finance management, production management, goods management, launch management, sales management, R&D management, institutional management), and subsystems of production system – all of that generates a whole group of managerial tasks that have to be solved together for each time period Tð Þ<sup>i</sup> ; the interrelation of

Work with a model structure means that we need to consider several subtasks related to forecasting parameters of the considered project [15] and to formalizing an optimization task

The examples of tasks that are considered in decision points can encompass the tasks of production and client analytics taking into account time factor, such as demand forecast and sales planning, volume planning, stock and procurement planning (including working life), equipment selection taking into account maintenance costs; these can be the tasks of optimizing stock work and minimizing the volumes of working assets, and obtaining optimal machine

<sup>j</sup> is the components or blocks of the model for time <sup>T</sup>ð Þ<sup>i</sup> ).

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79

that will be collected as a composition (the compatibility of input and output areas).

the model, investigate the object and look for the ways to improve management object).

ment maintenance periods, internal technological cycles etc.) [13].

employ methods to work with probabilistic models etc.).

mð Þ<sup>i</sup>

<sup>j</sup> ð Þ U; P; Π

Π is the vector of system parameters, mð Þ<sup>i</sup>

the tasks is demonstrated in Figure 2.

utilization and work force.

in a form of mathematical programming [16].

where w ¼ w1; w<sup>2</sup> f g ;…; w<sup>n</sup> is the projects' vector, Y is the action vector on each step, R is the outcome vector on each step, Pp is the vector of system states, Pa is the vector of system goal states, T is the vector of decision points.

The use of the model (1) is described by an undetermined algorithm [11] see Figure 1.

As a result, management task becomes more transparent. However, it opens new sub-tasks, i.e. to determine decision points, to define the set of indexes and their values for each stage of project implementation, to build a model of production system by implementing the projects (w) in order to define the vector of control actions Y.

At the same time, the more formalized is the description of tuple parts (1) (less ambiguity), the higher is the quality of management [according to system properties].

Figure 1. The algorithm to manage a production system that implements projects w.

Decision points can be defined in case if we know the set of controlled parameters [12], and have additional information that characterizes the production system that we manage (equipment maintenance periods, internal technological cycles etc.) [13].

In general, the model can be presented in a form of a tuple:

states, T is the vector of decision points.

78 Digital Transformation in Smart Manufacturing

(w) in order to define the vector of control actions Y.

higher is the quality of management [according to system properties].

Figure 1. The algorithm to manage a production system that implements projects w.

<sup>ψ</sup> <sup>¼</sup> <sup>Y</sup>; Pp; Pa; <sup>T</sup>;R; <sup>w</sup> (1)

where w ¼ w1; w<sup>2</sup> f g ;…; w<sup>n</sup> is the projects' vector, Y is the action vector on each step, R is the outcome vector on each step, Pp is the vector of system states, Pa is the vector of system goal

As a result, management task becomes more transparent. However, it opens new sub-tasks, i.e. to determine decision points, to define the set of indexes and their values for each stage of project implementation, to build a model of production system by implementing the projects

At the same time, the more formalized is the description of tuple parts (1) (less ambiguity), the

The use of the model (1) is described by an undetermined algorithm [11] see Figure 1.

The setting of management tasks taking into account time factor Tð Þ<sup>i</sup> leads to formalizing the models <sup>Y</sup>ð Þ<sup>i</sup> ! <sup>M</sup>ð Þ<sup>i</sup> <sup>A</sup>; <sup>Θ</sup>; <sup>T</sup>ð Þ <sup>i</sup>�<sup>1</sup> , Tð Þ<sup>i</sup> � �. The structure of the model sets formal interrelations between its parameters, and on each step the type of the model will depend on the managerial task that we consider (whether we forecast properties and behavior of the investigated management project; or when dealing with object management we select best actions by testing them on the model, investigate the object and look for the ways to improve management object).

The model itself can use both non-causal (component-oriented) and causal (block-oriented) modeling, and model components can set requirements to their development tool (for example, the possibility to 1) work with big data volumes set by time series 2) use the methods that are applied for incomplete data 3) solve tasks set in a form of mathematical programming 4) employ methods to work with probabilistic models etc.).

The specialization of models Yð Þ<sup>i</sup> brings the problem of choosing approaches and ways for formalization based on the set of already known approaches, ways, methods and models [14] that will be collected as a composition (the compatibility of input and output areas).

For the implementation of each project in the considered production system, the model formation that is presented in a general form is as follows f g <sup>R</sup>; <sup>w</sup>; <sup>A</sup>; <sup>Θ</sup> ! <sup>M</sup>ð Þ<sup>i</sup> <sup>A</sup>; <sup>Θ</sup>; <sup>T</sup>ð Þ <sup>i</sup>�<sup>1</sup> , Tð Þ<sup>i</sup> � � !

mð Þ<sup>i</sup> <sup>j</sup> ð Þ U; P; Π n o, where <sup>P</sup> is the vector of external parameters that exert impact on the system,

Π is the vector of system parameters, mð Þ<sup>i</sup> <sup>j</sup> is the components or blocks of the model for time <sup>T</sup>ð Þ<sup>i</sup> ).

Despite the apparent simplicity of the approach, underlying this approach is a necessity to work out managerial decisions taking into account different levels (institutional, managerial, technical) and management types (finance management, production management, goods management, launch management, sales management, R&D management, institutional management), and subsystems of production system – all of that generates a whole group of managerial tasks that have to be solved together for each time period Tð Þ<sup>i</sup> ; the interrelation of the tasks is demonstrated in Figure 2.

Work with a model structure means that we need to consider several subtasks related to forecasting parameters of the considered project [15] and to formalizing an optimization task in a form of mathematical programming [16].

The examples of tasks that are considered in decision points can encompass the tasks of production and client analytics taking into account time factor, such as demand forecast and sales planning, volume planning, stock and procurement planning (including working life), equipment selection taking into account maintenance costs; these can be the tasks of optimizing stock work and minimizing the volumes of working assets, and obtaining optimal machine utilization and work force.


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

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:

<sup>p</sup> costeqpgjð Þþ <sup>t</sup> <sup>P</sup>

• The minimization of costs for goods' storage, costjnyjnð Þþ t invjð Þt y1jnð Þþ t shipnijð Þt y2jnð Þt ! min, yjnðÞ¼ t y1jnð Þþ t y2jnð Þt , y2jnð Þt ≤ demjnð Þt , where y1jn-the number of items stored in

• The selection of suppliers taking into account that certain components can be reused,

j P

• The restriction related to delivery options of components and materials,

• Non-negativity restriction on the volumes of goods, orders etc., yjnð Þt , xijmð Þt , rnð Þt , omð Þt ,

<sup>j</sup> yjnð Þt ≤ demnð Þt , ∀n, t;

<sup>m</sup> priceim þ shipij þ invj

<sup>g</sup> eqpgjð Þt ≤ supmaxeqjpð Þt , ∀j, p, t;

� �xijm � <sup>P</sup>

<sup>m</sup> refcostjmref jm ! max.

<sup>n</sup> reqmnyjnðÞ¼ <sup>t</sup> <sup>P</sup>

<sup>j</sup> xijmð Þt ≤ supmaximð Þt sið Þt , ∀i, m, t,

<sup>n</sup> reqmnð Þt rnð Þt , ∀m,t;

<sup>j</sup> costjnð Þt � � P

<sup>n</sup> setdisnbdn

<sup>i</sup> xijmð Þþ t ref jmð Þt , ∀j, m, t,

P

<sup>j</sup> xijmð Þt ≥

<sup>m</sup> partmpgjð Þt shipmijð Þt

P p

<sup>m</sup> partmpgjð Þ<sup>t</sup> pricemið Þþ <sup>t</sup> <sup>P</sup>

� �

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<sup>j</sup> yjnð Þt

81

• Profit maximization (production planning for demand), sellnðÞ� <sup>t</sup> <sup>P</sup>

stock, y2jn is the number of items sent to consumer;

i P j P

om <sup>þ</sup> dispmdmÞ � <sup>P</sup>

that can be reused.

! max, ∀n;

! min, <sup>∀</sup>g, j;

P j P

�

P

P

costeqpgj � <sup>P</sup>

• Production cost minimization, P

<sup>n</sup> selln � costjn � �yin � <sup>P</sup>

<sup>m</sup> ðdisam

• Production capacity restriction, P

dmð Þt , ref jmð Þt ≥ 0, ∀j, n, i, m, t; • Demand volume restriction, P

supminimð Þt sið Þt , ∀i, m, t;

The tasks can be subject to different restrictions:

<sup>g</sup> partmpgjð Þt ≤ supmaxpartjmð Þt , ∀j, p, m, t;

• The description of technological process, P

<sup>j</sup> ref jmð Þþ <sup>t</sup> dmðÞ¼ <sup>t</sup> om, <sup>∀</sup>m, t, omðÞ¼ <sup>t</sup> <sup>P</sup>

• The restriction on the volume of orders, P

Figure 2. The interrelation of management levels and management tasks to be solved by using parameters and indicators for developing decision support models.

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 parameters and use forecast-based data.
