UDC 389.001

ON THE RATIONAL CONSTRUCTION OF METROLOGICAL LABORATORIES

PART II

E. N. RUZAEV, E. I. TSIMBALIST

(Presented by the scientific seminar of the Department of Radio Engineering)

Arrangement from Blocks of Separate Metrological Laboratories

Partial and integral optimality criteria for dividing a metrological complex into laboratories have been obtained. An algorithm has been developed for arranging two laboratories from blocks. An example illustrating this algorithm is given.
Illustrations: 2.

When dividing a metrological complex into separate laboratories, it is necessary to have an optimality criterion for such a division.

In general form, the criterion may be represented as a functional combining partial criteria of division, for which it is rational to use the following:

1) minimal duplication of reference measuring instruments in different laboratories — $\alpha$;

2) equal capacity of the RMI (their equality in all laboratories) — $\beta$;

3) equal capacity of subsets of parameters verified in each of the laboratories — $\gamma$;

4) equal verification time for the input flow of WMI in each of the laboratories — $\delta$;

5) equal cost of conducting verification in each of the laboratories — $\varkappa$.

The general criterion in this case will be written as

\[ W = F(\alpha, \beta, \gamma, \delta, \varkappa). \]

However, in a certain situation each of the partial criteria may prevail over all the others. Therefore, as the first step in optimizing the division of a metrological complex into laboratories, the solution of this problem using partial criteria will be used. To simplify the presentation of the material, we shall consider the question of division into two metrological laboratories (I, II).

1. Optimization of the Division According to Criterion $\alpha$

The objective function in this case will have the form

\[ W_1 = \varkappa \to \min,\quad \varkappa \in \mathrm{I},\ \mathrm{II}, \]

where $\varkappa$ is the number of duplicated RMI.

Let us consider the example (Fig. 1, part I of the article) according to criterion \(\alpha\); we note that the optimal partition will be the division of the complex into blocks by connected component. In this case \(W_1=\min=0\). The resulting laboratories can be represented in the form of two subsets:

\[ a)\quad \{AB\}\{CDEF\ GKM\}. \]

Less relative (rational) partitions are obtained by absorbing individual blocks (Fig. 4, part I of the article) into two laboratories. All blocks obtained when duplicating various MSI can be represented as sets:

\[ \{AB\}\{CD\}\{CE\}\{EF\}\{FGK\}\{KM\}. \]

By successive absorptions of subsets and their transformation into two subsets, the following partition variants can be obtained:

\[ \begin{aligned} b)&\quad \{A\ B\ F\ G\ K\}\ \{K\ M\ F\ E\ C\ D\};\\ c)&\quad \{F\ G\ K\ M\}\ \{A\ B\ C\ D\ E\ F\};\\ d)&\quad \{E\ F\ G\ K\}\ \{K\ M\ C\ E\ D\ A\ B\};\\ e)&\quad \{A\ B\ C\ D\ E\}\ \{E\ F\ G\ K\ M\};\\ f)&\quad \{A\ B\ C\ D\}\ \{C\ E\ F\ G\ K\ M\} \end{aligned} \]

and so on.

When considering variants \((b—f)\) according to criterion \(\alpha\), we are convinced that all these solutions are equivalent with respect to \(\alpha\),

\[ W_1=1, \]

except for solutions \(b\) and \(d\),

\[ W_1^b=W_1^d=2. \]

2. Optimization of the partition according to criterion \(\beta\)

The objective function for partitioning into two laboratories in this case will take the form

\[ W_2=|N_{\mathrm{I}}-N_{\mathrm{II}}|\to \min, \]

where \(N_{\mathrm{I}}\) and \(N_{\mathrm{II}}\) are the numbers of MSI in laboratories I and II, respectively. Let us consider variants \(a—f\) according to criterion \(\beta\):

\[ \begin{aligned} W_2^a&=|2-7|=5\\ W_2^b&=|5-6|=1\\ W_2^c&=|4-6|=2\\ W_2^d&=|4-7|=3\\ W_2^e&=|5-5|=0\\ W_2^f&=|4-6|=2. \end{aligned} \]

Thus, the optimal partition according to criterion \(\beta\) is solution \(e\), giving the following laboratories:

\[ \{A\ B\ C\ D\ E\}\ \{E\ F\ G\ K\ M\}. \]

Moreover,

\[ W_1^e=1. \]

3. Optimization of the partition according to criterion \(\gamma\)

The objective function for partitioning into two laboratories in this case will take the form

\[ W_3=|m_{\mathrm{I}}-m_{\mathrm{II}}|\to \min, \]

where \(m_{\mathrm{I}}\) and \(m_{\mathrm{II}}\) are the total numbers of parameters of the WMI verified respectively in laboratories I and II. The total number of parameters (including

and identical) WMI verified by the entire complex is equal to the number of edges of the König graph (i.e., to the total number of connections between WMI and SMI) for the given metrological-service system.

When partitioning into blocks by duplicating various reference measuring instruments, the total number of edges of all blocks is equal to the number of edges of the original König graph. Let us write down the number of graph edges that fell into the individual blocks.

Block 1 \(\{A B\}\) — 4
Block 2 \(\{C D\}\) — 3
Block 3 \(\{C E\}\) — 2
Block 4 \(\{E F\}\) — 2
Block 5 \(\{F G K\}\) — 4
Block 6 \(\{K M\}\) — 4

When partitioning into two laboratories, it is necessary, by mutual absorption of blocks, to obtain two blocks with the same number of edges. The total number of edges is 19. To obtain the optimal solution according to criterion \(\gamma\), it is necessary to form two blocks with approximately equal numbers of edges in each.

Let us analyze, according to criterion \(\gamma\), the solutions \(a\)—\(f\) given above:

a) \(\{A\ B\}\ \{C\ D\ E\ F\ G\ K\ M\}\)

\[ m\{A\ B\}=4=m_{\mathrm{I}} \]

\[ m_{\mathrm{II}}=m\{CD\}+m\{CE\}+m\{EF\}+m\{FGK\}+m\{KM\}= \]

\[ =3+2+2+4+4=15 \]

\[ W_3^a=|4-15|=11 \]

b) \(\{A\ B\ F\ G\ K\}\ \{K\ M\ F\ E\ C\ D\}\)

\[ m_{\mathrm{I}}=m\{AB\}+m\{F\ G\ K\}=4+4=8 \]

\[ m_{\mathrm{II}}=m\{KM\}+m\{FE\}+m\{CE\}+m\{CD\}=4+2+2+3=11 \]

\[ W_3^b=|8-11|=3 \]

c) \(\{FGKM\}\ \{A\ B\ C\ D\ E\ F\}\)

\[ m_{\mathrm{I}}=m\{FGK\}+m\{KM\}=4+4=8 \]

\[ m_{\mathrm{II}}=m\{AB\}+m\{CD\}+m\{CE\}+m\{EF\}=4+3+2+2=11 \]

\[ W_3^c=|8-11|=3 \]

d \(\{E\ F\ G\ K\}\ \{K\ M\ C\ E\ D\ A\ B\}\)

\[ m_{\mathrm{I}}=m\{EF\}+m\{FGK\}=2+4=6 \]

\[ m_{\mathrm{II}}=m\{KM\}+m\{CE\}+m\{CD\}+m\{AB\}=4+2+3+4=13 \]

\[ W_3^d=|6-13|=7 \]

e) \(\{A\ B\ C\ D\ E\}\ \{E\ F\ G\ K\ M\}\)

\[ m_{\mathrm{I}}=m\{AB\}+m\{CD\}+m\{CE\}=4+3+2=9 \]

\[ m_{\mathrm{II}}=m\{EF\}+m\{FGK\}+m\{KM\}=2+4+4=10 \]

\[ W_3^e=|9-10|=1 \]

etc.

Of all the variants analyzed according to criterion \(\gamma\), the most optimal is solution \(e\). In addition, solution \(e\) is optimal according to criterion \(\beta\) and rational according to \(\alpha\).

4. Optimization by the criterion \(\delta\)

The objective function when partitioning into two laboratories in this case will take the form

\[ W_4 = |T_{\Sigma I} - T_{\Sigma II}| \to \min , \]

where \(T_{\Sigma I}\) and \(T_{\Sigma II}\) are the total time for verification of RSI in laboratories I and II, respectively.

The total time for verification of RSI in a laboratory will depend on the verification principle used: sequential, parallel, or parallel-sequential.

For parallel verification of RSI,

\[ T_\Sigma = t_{\text{ver. max }j},\quad j \in \{1 \div n\}. \]

The total verification time is equal to the maximum verification time of an RSI among all \(n\) RSI submitted simultaneously for verification. Accordingly, \(n\) verification operators are required. For parallel-sequential verification of RSI,

\[ T_\Sigma = \sum_{i=1}^{n/n_1} t_{i\,\text{ver}}, \]

where \(n_1\) is the number of verification operators.

For sequential verification of RSI,

\[ T_\Sigma = \sum_{i=1}^{n} t_{i\,\text{ver}} . \]

In our example we shall restrict ourselves to considering sequential verification of both the parameters of the RSI and the RSI themselves.

In this case each edge of the graph will be assigned a weight equal to the verification time of the corresponding parameter of the RSI. If the verification times for each parameter are equal, i.e., if \(t_1 = t_2 = \ldots = t_{19}\), the given problem becomes completely analogous to the partitioning problem by the criterion \(\gamma\).

In the general case, considering the verification times of the parameters to be known, we place them (in conventional units) on the edges of the graph (Fig. 5). Let us write out the total verification times for the RSI in each of the blocks:

Block 1 \(\{A B\}\) — 4
Block 2 \(\{C D\}\) — 4
Block 3 \(\{C E\}\) — 4
Block 4 \(\{E F\}\) — 3
Block 5 \(\{F G K\}\) — 5
Block 6 \(\{K M\}\) — 4

The total time for verification of the entire unit park of input instruments is

\[ T_\Sigma = 24 \text{ units}. \]

Fig. 5

The optimal partition will be one that gives two laboratories, in each of which 12 units of time are required for verification of the RSI.

Let us consider, by the criterion \(\delta\), solution \(e\):

\[ \{A\ B\ C\ D\ E\}\ \{E\ F\ G\ K\ M\} \]

\[ T_{\Sigma \mathrm{I}}=T\{AB\}+T\{CD\}+T\{CE\}=4+4+4=12 \]

\[ T_{\Sigma \mathrm{II}}=T\{EF\}+T\{FGK\}+T\{KM\}=3+5+4=12 \]

Solution \(e\) is optimal according to criterion \(\delta\).

\[ W_4^e=\left|T_{\Sigma \mathrm{I}}-T_{\Sigma \mathrm{II}}\right|=|12-12|=0. \]

5. Optimization of the partition according to criterion \(\chi\)

The objective function when partitioning into two laboratories in this case will take the form

\[ W_5=\left|C_{\Sigma \mathrm{I}}-C_{\Sigma \mathrm{II}}\right|\to \min, \]

where \(C_{\Sigma \mathrm{I}}\) and \(C_{\Sigma \mathrm{II}}\) are the total cost of carrying out verification in each of the laboratories.

The cost of verifying one parameter \(j\) can be written as

\[ C_j=C_j^{1}\cdot t_j, \]

where \(C_j^{1}\) is the cost expended per unit time when verifying parameter \(j\);

\(t_j\) is the time for verifying parameter \(j\).

In the special case, if \(C_j=\mathrm{const}\) for \(j=1\div k\), then the partitioning problem according to criterion \(\chi\) is equivalent to the partitioning problem according to criterion \(\delta\), since \(C_j^{1}\) can be taken out and ultimately eliminated as a constant factor, for we are interested only in relative estimates. In the general case, assuming the cost coefficients are known, we place them (in conventional units) on the edges of the graph (Fig. 5).

Let us write out the total cost of verifying the measuring system in each of the blocks.

For example: \(C_1=C_A\cdot t_{A1}+C_A\cdot t_{A2}+C_B\cdot t_{B1}+C_B\cdot t_{B2}\), etc.

Block 1 \(\{AB\}\) — \(C_1=1\cdot1+1\cdot1+1\cdot1+1\cdot1=4\)

Block 2 \(\{CD\}\) — \(C_2=1\cdot2+2\cdot1+2\cdot1=6\)

Block 3 \(\{CE\}\) — \(C_3=1\cdot3+2\cdot1=5\)

Block 4 \(\{EF\}\) — \(C_4=2\cdot1+1\cdot2=4\)

Block 5 \(\{FGK\}\) — \(C_5=1\cdot1+2\cdot1+1\cdot2+1\cdot1=6\)

Block 6 \(\{KM\}\) — \(C_6=1\cdot1+2\cdot1+1\cdot1+2\cdot1=6\)

\[ C_{\Sigma}=C_1+C_2+C_3+C_4+C_5+C_6=4+6+5+4+6+6=31 \]

An optimal partition will be one that yields two laboratories, in each of which 15–16 cost units are spent on verification of the measuring system.

Let us consider solution \(e\) according to criterion \(\chi\).

\[ \{A\ B\ C\ D\ E\}\ \{E\ F\ G\ K\ M\} \]

\[ C_{\Sigma \mathrm{I}}=C\{AB\}+C\{CD\}+C\{CE\}=4+6+5=15 \]

\[ C_{\Sigma \mathrm{II}}=C\{EF\}+C\{FGK\}+C\{KM\}=4+6+6=16 \]

\[ W_5^e=\left|C_{\Sigma \mathrm{I}}-C_{\Sigma \mathrm{II}}\right|=|15-16|=1=W_{5\min} \]

Thus, solution \(e\) is optimal according to criterion \(\chi\).

6. Derivation of the generalized partition criterion

To obtain a generalized partition criterion, it is rational to use the generally accepted method of summing partial criteria with the corresponding weight coefficients, reflecting the rela-

considerable importance of this criterion over the others. In this case the generalized partition criterion can be written in the form

\[ W_{\Sigma}=P_{1}W_{1}+P_{2}W_{2}+P_{3}W_{3}+P_{4}W_{4}+P_{5}W_{5} =\sum_{i=1}^{5}P_iW_i. \]

The weighting coefficients \(P_1 \div P_5\) are determined as the result of expert analysis.

In the case of equal significance of all criteria,

\[ W_{\Sigma}=\sum_{i=1}^{5}W_i. \]

7. Algorithm for Composing Two Laboratories from Blocks

Step 1. By absorbing individual subsets (blocks), several variants of the partition are formed.

Step 2. Input of the initial data. Characteristics of the blocks:

a) number of OSI;
b) number of edges in each subgraph;
c) edge weights: \(t_j, C_j\).

Step 3. Computation of the values \(W_1 \div W_5\) for all possible partition variants.

Step 4. Determination of \(W_{\Sigma}\) for all partition variants.

Step 5. Selection of \(W_{\Sigma \min}\) from among all possible variants.

Step 6. Printing of the final optimal partition.

For our example there is no point in calculating all variants, since it is quite obvious that variant \(e\) is the most optimal, because according to the criteria \(\beta — W_2\), \(\gamma — W_3\), \(\delta — W_4\), \(\chi — W_5\) this variant is optimal, i.e., the best, and only according to criterion \(\alpha — W_1\) is it rational.

The final partition for our example is then

\[ \{A\,B\,C\,D\,E\}\{E\,F\,G\,K\,M\} \]

(Fig. 6).

Fig. 6