Keywords

1 Introduction

To meet the current needs of the consumer and manufacturer, a multilevel hierarchy of product quality indicators at the production stage requires identification of the nature of quality indicators. At the same time, the multilevel hierarchy of quality indicators in accordance with the concept of the product quality monitoring model [1] must meet the following requirements:

  • The behavior of the lower level should be limited to the requirements of the upper level.

  • The quality function (main level) should not include indicators aimed at ensuring quality (sublevels), but these indicators should characterize the level of output quality (should act as restrictions).

  • The hierarchy of quality indicators should include both qualitative and quantitative indicators.

  • Quality indicators must meet the requirements of standardization, comparability, representativeness, sensitivity at threshold values, and the absence of duplication of quality indicators at each level.

Indicators of the developed product quality along with the supporting information will allow regulating the quality of product flexibly. In particular, it will allow managing (in frames of Quality management) the enterprise resource planning (ERP) system and the production process management system. Meeting the Industry 4.0 requirements will allow optimizing production chain processes, such as logistics, design, production, operation, and after-sales service, in terms of time, resources, and quality loss [2,3,4].

2 A Mathematical Model for Assessing of the Product Quality

Prior to identifying product quality indicators, it is necessary to describe a mathematical model for evaluating the product quality level. The mathematical model for evaluating the quality level will be based on a two-level linear optimization model, in which the first level is the leader, and the second level is the follower.

The assessment of the quality level is determined from the following set-theoretic definition of the model:

$$\left\langle {Q,X,F_{i} ,Y_{i} } \right\rangle$$

where \(Q\) is the quality function (main level); \(X\) is the area for determining the numerical values of the quality function; \(F_{i}\) is the target functions (sublevels); and \(Y_{i}\) is the area for determining the values of the target function.

The finding of an optimal solution of the function (1) is carried out bottom-up: first, the optimal value of the sublevels \(F_{i} (Y^{i} )\) are found, then these values are substituted in (1), and values for the master level are found.

The optimization problem for the top level, taking into account the restrictions imposed under the levels, looks like this [5, 6]:

$$\mathop {\min }\limits_{x} \left\{ {^{{\begin{array}{*{20}c} {} \\ \end{array} }} Q[y(x),x]:G[y(x),x] \le 0,_{{\begin{array}{*{20}c} {} \\ \end{array} }} H[y(x),x] = 0,_{{\begin{array}{*{20}c} {} \\ \end{array} }} y(x) \in \psi (x)} \right\}$$
(1)

where \(y(x) = F_{i}\),\(F_{i} :y(x) \in \psi (x)\),\(\psi (x)\)—polyhedron, domain of constraints such that (\(Q:R^{n} \times R^{m} \to R,_{{\begin{array}{*{20}c} {} \\ \end{array} }} G:R^{n} \times R^{m} \to R^{k} ,_{{\begin{array}{*{20}c} {} \\ \end{array} }} H:R^{n} \times R^{m} \to R^{l}\) for \(k\) indexes, there are restrictions with the sign “\(\le\)”, and for \(l\) th indexes, there are restrictions with the sign “ = ”). The target functions for the decentralization task, based on the requirements for the quality indicators monitoring model, are as follows. \(Q(x,y_{1,2} )\) is the effectiveness of the monitoring model, \(F_{1} (x,y_{1,2} )\) is the cost function for the quality, and \(F_{2} (x,y_{1,2} )\) is a supplier management function. Variables x are such that \(x \in X \subset R^{n}\). The quality function is such that \(Q:X \times Y_{1} \times Y_{2} \to R\), where \(y_{{}}\) is quality criteria, so \(y_{i} \in Y_{i} \subset R^{mi}\), and \(F_{i} :X \times Y_{i} \to R\).

$$\begin{gathered} \mathop {\min }\limits_{x \in X} Q(x,y_{1} = y,y_{2} = z) = cx - d_{1} y + d_{2} z \hfill \\ Ax + B_{1} y_{1} + B_{2} y_{2} \le b_{1} \hfill \\ \mathop {\min }\limits_{{y_{i} \in Y}} F_{1} (x,y) = cx + d_{1} y \hfill \\ Ax + B_{1} y \le b^{/} \hfill \\ \mathop {\min }\limits_{{z_{i} \in Z}} F_{2} (x,z) = cx + d_{2} z \hfill \\ Ax + B_{2} z \le b^{//} . \hfill \\ \end{gathered}$$
(2)

The state of operation of the upper level is determined by the following output parameters (see Table 1).

Table 1. Output parameters of the status of the functioning of the top level.
Full size table
Table 2. The example of a task.
Full size table

3 Formalization of Unique Criteria for a Model for Assessing of Quality of Instrument-Making Products

To obtain an updated value of the proposed indicator, it is necessary to take into account the specifics of the production of instrumentation products, namely.

  • structural complexity of the product after performing a certain technological operation is \(X_{1}\);

  • the shareof components with deviations by deviation is \(X_{2}\);

  • the shareof components with marriage certificates is \(X_{3}\);

  • the shareof components with acts of noncompliance is \(X_{4}\);

  • the shareof purchased items included in components (component parts of products) is \(X_{5}\).

Structural complexity of the product \(x_{1}\) is defined as the ratio of the complexity of the product at a certain operation (assigned index p) to the structural complexity of the product at the output (on the final operation, assigned index f):

$$x_{1} = \frac{{C_{p} }}{{C_{f} }} = \frac{{m_{p} }}{{n_{p} (n - 1)}} \times \frac{{n_{f} (n - 1)}}{{m_{f} }}.$$
(3)

where \(C_{p}\) is the complexity of products for a particular operation; \(C_{f}\) is the complexity of the product output; \(m\) is the number of elements in the product, and \(n\) is the number of connections in the product.

Information on \(X_{2}\), \(X_{3}\), and \(X_{4}\) will be recorded by counting cases of registration of permits for rejection, acts of marriage, and acts of noncompliance. The share of purchased items \(X_{5}\) will be defined as the ratio of purchased items to the total number of used items in the product (design).

In the indicator “the degree of effectiveness of the developed preventive measures (for the previous period)”, the following criteria should be taken into account:

  • the share of corrected (simple) technological operations is \(Y_{1}\);

  • the share of corrected (specially responsible) technological operations is \(Y_{2}\);

  • the share of the revised documents of the quality management system is \(Y_{3}\);

  • the percentage of corrected critical blocks in the product is \(Y_{4}\);

  • the percentage of corrected (noncritical) blocks in the product is \(Y_{5}\);

  • the percentage of corrected (not visible to the consumer) functions of the product is \(Y_{5}\).

The indicator “degree of efficiency from newly introduced technologies” determines how successful the investment was, and should be determined by the following criteria:

  • the share of the cost of developing a technological operation is \(Z_{1}\);

  • the share of expenses for the development of quality management system documentation is \(Z_{2}\);

  • the share of the cost of change of supplier is \(Z_{3}\);

  • the share of expenditure on adjustment of the design is \(Z_{4}\);

  • the share of costs for changing the functioning of the product is \(Z_{5}\).

These elements of a cost must be taken into account when evaluating the single criterion “degree of efficiency from newly technologies introduced”.

Let’s set a linguistic variable to determine the level of output of suitable products. A linguistic variable is defined as follows:

$$\left\langle {X_{i} ,T(X_{i} ),U,G,M} \right\rangle ,$$

where \(X_{i}\) is a name of the variable, \(T(X_{i} )\) is a term-set of the variable \(X_{i}\), \(U\) is the universal set, \(G\) is a syntactic rule, and \(M\) is a semantic rule.

The term set \(T(X_{i} )\) for the linguistic variables “level of output of suitable products”, “degree of effectiveness of developed warning measures (for the previous period)”, and “degree of efficiency from newly introduced technologies” will be low value, average value, and high value. The universal set \(U = [0,1]\), and \(G:M = [1 - u_{t} (u),(u_{t} (u))^{2} ]\).

The following scale will be used in the rule database for the place of output value names:

  • low value \(x \le 0,25\);

  • not the average value \(0,25 < x < 0,45\);

  • average \(0,45 \le x \prec 0,6\);

  • low value \(0,6 \le x < 0,75\);

  • high value \(0,75 \le x < 0,85\);

  • very high value \(0,85 \le x\).

The following rating scale will be used for the place of names of input values:

  • low value \(x < 0,4\);

  • average \(0,4 \le x < 0,8\);

  • high value \(0,7 \le x\).

The following functions will be used for input information:

  • Z-shaped accessory function for \(x < 0,4\);

  • Triangular accessory function for \(0,4 \le x < 0,8\);

  • S-shaped accessory function for \(0,7 \le x\).

To get a numerical expression of the quality indicator, the relative importance coefficients for each criterion must be determined. A mathematical model for determining relative importance coefficients is described as

$$\left\langle {f_{\Omega } ,_{{\begin{array}{*{20}c} {} \\ \end{array} }} K(X,Y,Z)_{1}^{{n_{1} }} ,_{{\begin{array}{*{20}c} {} \\ \end{array} }} ...,_{{\begin{array}{*{20}c} {} \\ \end{array} }} K(X,Y,Z)_{m}^{1} ,_{{\begin{array}{*{20}c} {} \\ \end{array} }} ...,_{{\begin{array}{*{20}c} {} \\ \end{array} }} K(X,Y,Z)_{m}^{{n_{m} }} ,_{{\begin{array}{*{20}c} {} \\ \end{array} }} R_{N} } \right\rangle$$

where \(f_{\Omega }\) is the area of determining of the output values of relative importance coefficients, \(K(X,Y,Z)_{1,...,m}^{{1,...,n_{m} }} = \left\langle {K(X)_{1,...,m}^{{1,...,n_{m} }} ,K(Y)_{1,...,m}^{{1,...,n_{m} }} ,K(Z)_{1,...,m}^{{1,...,n_{m} }} } \right\rangle\), criteria \(K_{i}\) is such that each corresponds to a set \(n_{i}\) (number of responses) of equally important criteria \(K_{i}^{1} ,...,K_{i}^{{n_{i} }}\) and the non-strict preference ratio \(R_{N}\).

The quantitative importance information \(\vartheta\) included in \(y(x )\) contains information about the degree of superiority of i, the criterion over other \(j\) criteria \(i \succ^{h} j\), where h answers the question “how many times one criterion is more important than the other” [7, 8].

The implication operation is performed as \(\mu_{A\Lambda B\Lambda C}\) \((x,y,z) = \min\) \(\{ {\mu_{A} (x),\mu_{B} (y),\mu_{C} (z)} \}\). The Z-shaped function will be found for \(a = 0\) and \(b = 1\). The S-shaped function will be found for \(a = 0\) and \(b = 1\). The triangular function will be found for \(a = 0\), \(b = 0,5\), and \(c = 1\). After finding the fuzzy value, the truth of the statement will be found by the Zane function, where \(a\) is an upper bound of \(i\) th scale interval for output values, presented in the previous sections. For numerical formalization of the above criteria, it is necessary to determine the cost management model and its mathematical form. There are many models of cost management; the main models given in [9, 10] are the following: the system “standard-cost”, target-costing method, the method of direct costing.

The most appropriate model for cost management is the model given in [11].

$$PR = \sum\limits_{i = 1}^{n} {P_{i} Q_{i} - \left( {\sum\limits_{i = 1}^{n} {vc_{i} } Q_{i} + FC} \right)}$$
(4)

where \(n\) is the number of types of products, \(P\) is the price of products of \(i\)th type, \(Q_{i}\) is a number of products of \(i\)th type, and \(vc_{i}\) are variable costs per unit of production of products of \(i\)th type, variable costs are \({\text{vc }} = {\text{ U}}_{{{\text{VC}}}} {\text{ P}}_{{uVC}} ,\) where \(U_{VC}\) is the specific consumption of the resource, and \({\text{P}}_{{{\text{uVC}}}}\) is the price per unit.

To apply the model (4), the individual criteria for each particular criterion of the quality cost function must be defined. The costs of quality assessment are characterized by the following single criteria [12, 13]:

  • The share of expenditure on critical functions that are visible to the consumer and the total costs is \(X^{\prime}_{1}\);

  • The share of costs for noncritical functions visible to the consumer to total costs is \(X^{\prime}_{2}\);

  • The share of costs for noncritical functions that are not visible to the consumer is \(X^{\prime}_{3}\).

The costs of preventing nonconformities are characterized by the following single criteria:

  • The share of expenditure on critical functions that are visible to the consumer and the total costs is \(Y^{\prime}_{1}\);

  • The share of costs for noncritical functions visible to the consumer to total costs is \(Y^{\prime}_{2}\);

  • The share of costs for noncritical functions that are not visible to the consumer is \(Y^{\prime}_{3}\).

The costs of eliminating internal inconsistencies are characterized by the following single criteria:

  • The share of costs eliminated by working with the supplier to the total cost of resolving internal inconsistencies is \(Z^{\prime}_{1}\);

  • The share of costs included in the manufacturing technology to the total cost of eliminating internal inconsistencies is \(Z^{\prime}_{2}\).

In the case of the listed criteria in the model (4), it is necessary to substitute the argument from \(X^{\prime}\),\(Y^{\prime}\),\(Z^{\prime}\) in the specific resource consumption and in the variable costs \(vc_{i}\), as follows:\(vc_{i} (X^{\prime})\),\(vc_{i} (Y^{\prime})\),\(vc_{i} (Z^{\prime})\), \(U_{VC} (X^{\prime})\), \(U_{VC} (Y^{\prime})\), and \(U_{VC} (Z^{\prime})\).

A cost-ratio model approach will be used to calculate single criteria for the supplier management function. An analysis of this approach is presented in [14, 15]. The private criteria for the supplier management function are based on the following unique criteria [16,17,18]:

  • the share of suppliers with a critical component of the total number of suppliers is \(X^{\prime\prime}_{1}\), \(Y^{\prime\prime}_{1}\), \(Z^{\prime\prime}_{1}\);

  • the proportion of outsourcing on the total number of suppliers is \(X^{\prime\prime}_{2}\), \(Y^{\prime\prime}_{2}\), \(Z^{\prime\prime}_{2}\);

  • the share of suppliers with imported products from the total number of suppliers is \(X^{\prime\prime}_{3}\), \(Y^{\prime\prime}_{3}\), \(Z^{\prime\prime}_{3}\);

  • the share of suppliers that make up 15% of the total cost of purchasing components to the total number of suppliers (products with high cost) is \(X^{\prime\prime}_{4}\), \(Y^{\prime\prime}_{4}\), \(Z^{\prime\prime}_{4}\).

The timeliness of delivery is based on the formula [19, 20]:

$$b^{\prime\prime}_{i1} z_{1} = \left[ {\frac{{\sum\nolimits_{i = 1}^{n} {z_{1} (t)_{i} } }}{m}} \right] \times \frac{1}{{z(t)_{b} }} \times 1^{{1 - d_{1} }} ,$$
(5)

where \(z_{1} (t)_{i}\) is the time needed to eliminate defects for customer satisfaction, \(m\) is a number of reference points, \(z_{1} (t)_{b}\) is the time needed to eliminate defects for customer satisfaction under the contract, and \(d_{1}\) is the ratio of the amount of the cost of components and the cost of applying them to their destination to the cost of the flaw detector.

Criterion \(d_{2} z_{2}\) will be found by the formula:

$$b^{\prime\prime}_{i2} z_{2} = \left[ {\frac{{\sum\limits_{i = 1}^{n} {z_{2} (t)_{i} } }}{n}} \right] \times \frac{1}{{z_{2} (t)_{b} }} \times 1^{{1 - d_{2} }} ,$$
(6)

where \(z_{2} (t)_{i}\) is time points for delivering components, \(n\) is the number of reference points, \(z_{2} (t)_{b}\) is the time to deliver components under the contract, and \(d_{2}\) is the ratio of the amount of the cost of components and the cost of applying them to their destination to the cost of the flaw detector.

Criterion \(d_{3} z_{3}\) will found by the formula:

$$b^{\prime\prime}_{i3} z_{3} = z_{3}$$
(7)

where \(z_{3}\) is the relation of closed questions to the total number of questions.

The term set \(T(X_{i} )\) for the linguistic variables “level of output of suitable products”, “degree of effectiveness of developed warning measures (for the previous period)” and “degree of efficiency from newly introduced technologies” will be low value, average value, and high value. The universal set is \(U = [0,1]\), and \(G:M = [1 - u_{t} (u),(u_{t} (u))^{2} ]\). The following scale will be used in the rule database for the place of output value names:

  • low value \(x \le 0,25\);

  • not the average value \(0,25 < x < 0,45\);

  • average \(0,45 \le x < 0,6\);

  • low value \(0,6 \le x < 0,75\);

  • high value \(0,75 \le x < 0,85\);

  • very high value \(0,85 \le x\).

The following rating scale will be used for the place of names of input values:

  • low value \(x < 0,4\);

  • average \(0,4 \le x < 0,8\);

  • high value \(0,7 \le x\).

The following functions will be used for input information:

  • Z-shaped accessory function for \(x < 0,4\);

  • Triangular accessory function for \(0,4 \le x < 0,8\);

  • S-shaped accessory function for \(0,7 \le x\).

4 Calculating the Quality Level Using a Two-Level Optimization Model

The example of a task is listed in Table 2.

More detailed information about two-level programming could be found in [8]. Based on the results of the implementation of the simplex method, the quality level of the flaw detector at the production stage was determined as \(Q(x,y,z) = 0,67\).