2. Informal description

Let us consider a company, which consists of director, three departments, and one separate laboratory. This fact may be simply represented as follows:

$$
\mathbf{1}\text{company} \to \mathbf{1} \cdot \text{director}, \mathbf{3} \cdot \text{department}, \mathbf{1} \cdot \text{laboratory}.\tag{1}
$$

In this notation, a whole structure of the company, detailed to employee positions, may be described in such a way:

$$\begin{array}{l}\text{department} \rightarrow \texttt{1} \cdot \texttt{head-dependent}, \texttt{3} \cdot \texttt{labortory},\\\text{labortory} \rightarrow \texttt{1} \cdot \texttt{head-latency}, \texttt{2} \cdot \texttt{analyst}, \texttt{3} \cdot \texttt{assistant}.\end{array} \tag{2}$$

This set of constructions is of the form:

$$a \to n\_1 \cdot a\_1, \dots, n\_m \cdot a\_m \tag{3}$$

describes following set, created by multiplying and summarizing quantities of identical positions:

f1 � director; 3 � head � department; 10 � head � laboratory; 20 � analyst; 30 � assistantg,

where ni � ai means there are ni positions of type ai in this company.

director ! 10000 � eur,

analyst ! 1500 � eur, assistant ! 500 � eur:

represented in the same unified manner:

DOI: http://dx.doi.org/10.5772/intechopen.81698

employees' provision a month.

department does not exceed five):

follows:

71

Let us join to the company structure knowledge about employees' month salary,

Multiset-Based Knowledge Representation for the Assessment and Optimization of Large-Scale…

head‐department ! <sup>5000</sup> � eur, head‐laboratory ! <sup>3000</sup> � eur,

After applying to the joined set of constructions just the same multiplyingsummarizing procedure, we may obtain resulting set containing the only element {100,000�eur}, which defines company's total financial resource, necessary for

Presented knowledge representation concerns systems analysis, that is, obtaining integral parameters of the system given its structure and local parameters. Consider more sophisticated task-relating systems design and concerning development of company structure given its integral parameters. Goal is to determine rational quantity of departments and laboratories in the department, as well as quantities of analysts and assistants in one laboratory. Total salary is no more than 120,000 eur, quantity of analysts in one laboratory may be from 1 to 3, while corresponding quantity of assistants may be from 2 to 6. Total quantity of employees must be maximal. There may be three different variants of company structure: (1) three departments and one laboratory; (2) two departments and three laboratories; and (3) four departments. Corresponding set of constructions is as

> company ! 1 � director, 3 � department, 1 � laboratory, (6) company ! 1 � director, 2 � department, 1 � laboratory, (7)

department ! <sup>1</sup> � head‐department, m � laboratory, (9) laboratory ! <sup>1</sup> � head‐laboratory, n � analyst, l � assistant: (10)

Constructions, defining employees'salary, and other aforementioned restrictions are as follows (for definiteness, let us take that quantity of laboratories in one

company ! 1 � director, 4 � department, (8)

director ! 1 � employee, 10000 � eur, (11)

employee ¼ max, (16) eur ≤120000, (17) 1≤ m ≤5, (18) 1≤ n≤3, (19) 1≤l ≤6: (20)

head‐department ! <sup>1</sup> � employee, <sup>5000</sup> � eur, (12) head‐laboratory ! <sup>1</sup> � employee, <sup>3000</sup> � eur, (13) analyst ! 1 � employee, 1500 � eur, (14) assistant ! 1 � employee, 500 � eur, (15)

(5)

Multiset-Based Knowledge Representation for the Assessment and Optimization of Large-Scale… DOI: http://dx.doi.org/10.5772/intechopen.81698

where ni � ai means there are ni positions of type ai in this company.

Let us join to the company structure knowledge about employees' month salary, represented in the same unified manner:

$$\begin{aligned} &direct \rightarrow 10000 \cdot \textit{eur}, \\ &head \cdot \textit{department} \rightarrow 5000 \cdot \textit{eur}, \\ &head \cdot \textit{labortory} \rightarrow 3000 \cdot \textit{eur}, \\ &\andup \textit{st} \rightarrow \textbf{1500} \cdot \textit{eur}, \\ &\to 3 \textit{soistant} \rightarrow 500 \cdot \textit{eur}. \end{aligned} \tag{5}$$

After applying to the joined set of constructions just the same multiplyingsummarizing procedure, we may obtain resulting set containing the only element {100,000�eur}, which defines company's total financial resource, necessary for employees' provision a month.

Presented knowledge representation concerns systems analysis, that is, obtaining integral parameters of the system given its structure and local parameters.

Consider more sophisticated task-relating systems design and concerning development of company structure given its integral parameters. Goal is to determine rational quantity of departments and laboratories in the department, as well as quantities of analysts and assistants in one laboratory. Total salary is no more than 120,000 eur, quantity of analysts in one laboratory may be from 1 to 3, while corresponding quantity of assistants may be from 2 to 6. Total quantity of employees must be maximal. There may be three different variants of company structure: (1) three departments and one laboratory; (2) two departments and three laboratories; and (3) four departments. Corresponding set of constructions is as follows:

$$company \to \mathbf{1} \cdot director, \mathbf{3} \cdot department, \mathbf{1} \cdot laboratory,\tag{6}$$

$$company \to \mathbf{1} \cdot \text{direct}; \mathbf{2} \cdot \text{department}, \mathbf{1} \cdot \text{laborator} \mathbf{y},\tag{7}$$

$$company \to 1 \cdot director, 4 \cdot department,\tag{8}$$

$$
\begin{array}{c}
\text{department} \rightarrow \texttt{1} \cdot \textit{head-department}, \, m \cdot \textit{laborator} \chi, \\
\text{.} \; . \; . \; . \; . \; .
\end{array}
\tag{9}
$$

$$\text{laboratory} \to \mathbf{1} \cdot \text{head-laboratory}, \, n \cdot \text{and} \, \text{st}, \, l \cdot \text{assistant.} \tag{10}$$

Constructions, defining employees'salary, and other aforementioned restrictions are as follows (for definiteness, let us take that quantity of laboratories in one department does not exceed five):

$$direction \rightarrow \mathbf{1} \cdot employment, \mathbf{10000} \cdot err,\tag{11}$$

$$
\begin{array}{ccc}
\text{head-department} \rightarrow \mathbf{1} \cdot \textit{employee}, & \mathbf{5000} \cdot \textit{erur}, & \text{(12)} \\
\cdot & \cdot & \cdot
\end{array}
$$

head‐laboratory ! <sup>1</sup> � employee, <sup>3000</sup> � eur, (13)
