Introduction. The sustainability and efficiency of an industrial enterprise's production activities are largely determined by the quality of its resource provision and the rhythm of material supplies and product output [1]. In particular, the feasibility of using production organization methods such as just-in-time and the Kanban system depends on these factors [2]. Therefore, close attention should be paid to the assessment of these parameters, which necessitates the use of a specific set of indicators [3]. It should be noted that such indicators can also be used to assess the performance of the departments responsible for material supply and production process management [4]. At the same time, it should be kept in mind that the actual nomenclature of supplies and the range of current needs of an enterprise (or its subdivision, for example, a service department responsible for the maintenance of production equipment) do not generally coincide over a given period — neither in terms of the composition of items nor in the quantity of material resources of each type. This discrepancy is explained by the following reasons:

time lag between placing a purchase order for material resources and their actual delivery. By the time the resources are received, the enterprise's (or subdivision's) requirements for material resources may have changed — both in terms of nomenclature and the quantity of specific resource types;

organization of material and technical support (MTS) for the enterprise's subdivisions based solely on standards or pre-set plans, without taking current needs into account;

errors and failures in the performance of the enterprise's MTS department and its external suppliers (delivery failures and delays, deficiency, etc.). Since this department interacts with external suppliers, its responsibilities also include substandard work;

restrictions on the supply of foreign resources amid the current geopolitical situation (as experience shows, even paid orders may be suspended or frozen) [5].

This implies that the degree of alignment between actual deliveries and current needs serves as a basis for assessing both the quality of material resource provision to the enterprise and the performance quality of its MTS department (in particular, its ability to predict the resource needs of departments regardless of their requests and take these forecasts into account when placing orders with suppliers, as well as monitor the fulfillment of suppliers' obligations). Still to date, the issues arising in quantitatively assessing this conformance have been insufficiently studied. It is clear that upon receiving an order from a supplier, the enterprise's MTS checks the quality and quantity of the delivered materials (relevant regulations and standards exist for this purpose). However, this check is performed on a per-item basis, assessing only the specific resource within a given delivery. The author believes that in these situations, it is necessary to use a comprehensive methodology that will allow for an overall assessment of the degree to which supplies meet the enterprise's needs.

To assess the rhythm of production and supply, the rhythmicity index is most often used, determined by various methods [3], which also have certain shortcomings. For example, when calculating it, the impact of excess production is generally not taken into account [6]. This reduces the value of the rhythmicity index for assessing the quality of an enterprise's logistics (production) activities and, consequently, for making management decisions. These problems indicate the need to improve existing methods for determining the rhythm of production.

Thus, given the above, it can be argued that industrial enterprises may currently face challenges in assessing the extent to which actual material supplies correspond to current needs and in determining the level of production efficiency. This fact underscores the relevance of developing a scientific and methodological framework for a more comprehensive assessment of the logistics and production performance of industrial enterprises.

It should be noted that, at present, research addressing the aforementioned problems is virtually nonexistent. The publications known to the author on the organization of logistics and production processes can be grouped into four main categories.

The first category examines the problems of assessing the efficiency of logistics and production activities. It can be divided into two main areas. Within the first, new methods for assessing the efficiency of logistics and production activities are being developed (usually either by improving existing methods through incorporating additional parameters, such as production flexibility [7] or environmental friendliness [8], or by using complex mathematical apparatus for modeling production and logistics processes [9]). The second area is represented by research on the systematization and critical analysis of existing methods for assessing the production and logistics activities of companies [10]. However, none of these areas examines the issues of improving the methods for calculating indicators of the regularity and rhythm of deliveries and production, as well as the development of methods for assessing the conformance of actual and planned deliveries.

The second category of publications is represented by studies on the efficiency of production organization within an enterprise or even a separate division (or production site) [11], including the use of modern information technologies [12][13]. The works related to this category touch upon the problems of assessing the regularity and rhythm of production activities [14] (including the identification of factors that affect the rhythm of product release [15]). However, these problems are solved on the basis of already existing indicators and methods [16]. The task of improving the methods for assessing the rhythm of supplies and production is not set. The problems of assessing the conformance of actual supplies to current needs are also not addressed.

The third category includes supply chain management. Within this category, strategic and tactical research areas can be distinguished. The tactical aspect is primarily concerned with improving the efficiency of current supply chain operations [17]. This task requires metrics for its assessment [18], but these do not include indicators of supply conformance with the current needs of the enterprise, nor the rhythm of production. The focus here is primarily on meeting delivery deadlines [19] and the sustainability of supply chains [20]. Studies related to the strategic direction analyze the transformation of supply chains with the aim of providing their long-term sustainability [21]. Such transformation, specifically, can be realized through the introduction of digital tools [22][23] (including digital twin technologies [24], artificial intelligence [25], and blockchain [26]). The active use of environmentally friendly (green) technologies is also of great importance [27]. It is worth emphasizing that the use of digital and energy-saving technologies and alternative energy is not a distinctive feature of the development of supply chains. It corresponds to the global trend of economic systems development in the context of the digital technological order [28–30]. Finally, in the current situation, experts are paying much attention to the problems of providing the sustainability of global supply chains in the context of geopolitical instability [31]. The issues of developing methods for assessing the consistency of supplies and their conformance with current needs in this area are not presented.

The fourth category includes publications related to the development of integrated information and analytical systems [32] and algorithms [33] for managing the production activities of an industrial enterprise. Their authors formulate lists of characteristics of the company's economic activity [33] on whose basis management should be built (the rhythm of production can be included in these lists [34]). However, as a rule, they do not offer methods for calculating indicators for assessing these characteristics and do not provide recommendations for improving existing methods (since we are talking about an integrated management system, and not about individual indicators). Conformance of actual supplies to current needs is not included in this list.

The author was able to find a single study that attempted to assess the conformance of resource supplies with the current needs of the customer [35]. It is important to note that this study pertains to the organization of supplies within the framework of the logistics system of the Armed Forces of the Russian Federation, where such conformance is obviously of strategic importance. The results obtained are of undoubted interest. However, the selected assessment methodology, in the author's opinion, does not allow for a full determination of the conformance of supplies with needs.

Thus, it can be concluded that there is a gap in knowledge in the scientific literature regarding the conformance of material supplies with the needs of enterprises, which determines the objective of this study — the development of new methods for calculating the degree of conformance of actual supplies of material resources with the current needs of enterprises, as well as determining the assessment of the level of rhythm of both supplies and product output.

Materials and Methods. The following research methods and materials were used in the study:

a descriptive literature review to identify the current state of research in assessing the degree to which effective resource supplies match the current needs of an enterprise. For this purpose, the author consulted publications in scientific journals included in the scientometric databases of the Russian Science Citation Index (RSCI) and Scopus, as well as seminal works on this topic that laid out the basic approaches to developing relevant indicators [36]. The study utilized a general scientific method of analysis and synthesis;

mathematical modeling methods (including the use of set theory, primarily binary measures of set similarity [37–39]).

The theoretical basis of the research was the theory of production organization, primarily the methods developed within its framework for assessing the rhythm and consistency of production output [6].

Research Results. Assessing the conformance of actual resource supplies to the current needs of an industrial enterprise.

To determine such an estimate, the following notations were introduced:

F — a set of resources supplied by departments responsible for the logistics of the enterprise; D — numerous types of resources needed by the enterprise (division).

Taking into account the above, the total potential nomenclature of supplies M = F ∪ D (which includes both the resources actually supplied by the MTS of the enterprise, and those types of material resources that are needed by the company, but are not included in the supplies) consists of N types of material resources:

where n — cardinality of the corresponding set.

Value N can be described as the total range of supplies and demands.

Let Fi be the quantity of supplies of the i-th type of material resources performed during a given period by the departments responsible for the company's logistics (i ∈ 1, 2, …, N), and Di be the quantity of the enterprise's (division's) actual demands for resources of the i-th type over the same period.

It is obvious that the degree of correspondence between actual supplies and current needs can be assessed both in terms of the range of supplies and in terms of the quantity of resources of each type. To assess the correspondence of the range of supplies, one of the numerous currently available indicators of the degree of similarity between sets can be used [40]. One such measure of similarity is, in particular, the Jaccard index J [37][38]. The formula for calculating it, considering the previously introduced designations, will take the following form:

(1)

It is evident that

When J = 1, the range of actual supplies completely matches the range of current needs, while when J = 0, there is a complete mismatch between these ranges.

To assess the degree of mismatch between the actual range of supplies and current needs, the range mismatch indicator JN can be used:

(2)

where Q = F △ D — a set of resource types, whose actual supply does not correspond to the current needs of the enterprise.

Obviously, when JN = 1, the range of actual supplies does not completely coincide with the range of current needs, whereas when JN = 0, there are no deviations in composition between these ranges.

However, the condition J = 1 is not enough to determine the correspondence of actual supplies to current needs (and to evaluate the work of the enterprise's logistics departments as efficient), since in this case there may be deviations between the actual volume of supplies of resources and the current needs for these resources. Thus, it is required to take into account the magnitude of the needs and supplies of specific types of resources.

Nevertheless, in the author's view, from the perspective of the MTS departments' ability to meet the enterprise's needs, nomenclature mismatches may be of greater importance than quantity mismatches for individual resource types. A nomenclature discrepancy suggests that the MTS department is either unfamiliar with the enterprise's actual requirements (or unwilling to account for them), or that certain items are simply unavailable on the market (meaning that resolving such issues may involve substantial difficulties). By contrast, a quantity discrepancy for specific resource types points to isolated supply problems, which, as a rule, can be addressed relatively quickly.

To determine the degree of conformance between actual supplies and current needs for all types of material resources, an alternative version of the Jaccard index JA for descriptive sets (known as the Ruzicka coefficient [41][42]) can be also used. It is calculated using the following formula [42]:

(3)

However, as formula (3) shows, the disadvantage of this alternative Jaccard index is that it does not take into account the qualitative differences between the elements of the sets. In other words, units of resources of different types are considered equivalent to each other and are taken into account together (in particular, it is not considered that different types of resources may have different dimensions), which distorts the management meaning of the resulting indicator. This problem is also characteristic of other binary measures of set similarity (Sørensen coefficient, etc.) [39]. Furthermore, the Jaccard index may distort the conformance assessment due to the fact that elements with higher values have a greater impact on it than elements with lower values. This can be illustrated with the following simple example. Suppose an industrial enterprise orders two types of material resources. The order for the first type is 500 units (say, pieces), and for the second type — 4 units (say, tons). Effective deliveries of the first type of resource are equal to 500 units (pieces), and of the second type — 3 units (tons). Then the value of the alternative Jaccard index JA, calculated using formula (3), will be equal to:

Formally, this is a rather high value, but the enterprise’s need for the second type of resource is satisfied only by 75%, which formula (3) does not take into account. In fact, a sufficient number of “pieces” partially compensated for the shortage of “tons”. The result obtained is clearly devoid of any meaningful value for production and logistics management — neither quantitatively (i.e., with respect to the numerical value of the indicator) nor qualitatively (i.e., in terms of the feasibility of summing tons and pieces). Thus, in the case of significant differences between the quantities of deliveries of various types of resources measured in different units, the use of formula (3) may result in an inaccurate assessment of conformance. This may lead to incorrect management decisions [43] (in particular, within the framework of the above hypothetical example, the obtained sufficiently high value of the JA indicator most likely does not determine the need to improve the quality of the organization of the enterprise's logistics).

It should be emphasized that the methodology for assessing the conformance of supplies and needs proposed in [35] is based precisely on formula (3). For this reason, the author believes that this methodology does not fully reflect this conformance.

Therefore, to assess the degree of similarity between the composition of actual supplies and current needs for resources of the i-th type, the author proposes using the conformance indicator ICi:

(4)

Formula (4) can be rewritten as:

(5)

where IDi — coefficient of discrepancy between supplies and needs for the i-th type of material resources:

where sign (x) — function “sign”:

Formulas (4) and (5) are equivalent, but formula (5) is convenient in that it explicitly presents the nature of the discrepancy between current needs and actual supplies, which is described using the discrepancy coefficient IDi.

Formula (4) indicates the undesirability of supplying material resources in excess of current needs, since in this case, the enterprise may not be able to use them for their intended purpose, but at the same time resources will be diverted to storage.

It is obvious that

In this case, when ICi = 0, there is a complete discrepancy between actual supplies and current needs, whereas when ICi = 1, we can talk about their complete coincidence. In particular,

(6)

The author proposes to calculate the integral conformance indicator TIC for N types of material assets using the following formula:

(7)

Unlike formula (3), formula (7) takes into account units of resources of various types separately. At the same time, considering condition (6), the integral conformance indicator TIC takes into account not only the discrepancy between the current needs and actual supplies, but also the mismatch in their range. For the considered conditional example, the value of the integral conformance indicator TIC will be equal to:

In other words, for this hypothetical example, TIC < JA. Thus, formula (7) allows for more accurate consideration of both the presence of different types of resources in supplies and deviations between actual supplies and current needs, regardless of the magnitude of these needs, and, as a result, enables more efficient management decisions.

Finally, a weighted integral conformance indicator, WTIC, can also be proposed:

(8)

where Wi — weight assigned to the i-th type of material resources (reflecting their importance for the enterprise's production activities, storage complexity, market availability, cost, etc.). The weights satisfy the following condition:

However, using the WTIC indicator can be challenging due to the complexity of determining weights (specifically, because the same resource may have different significance for the enterprise at different periods of time, making the process of determining weights dynamic). Nevertheless, the following approaches can be used to set weight values for a definite period, whose duration depends on the specific enterprise's activities and the level of stability of its external environment:

expert assessment method;

hierarchy analysis method [44];

Fishburne progression (its classic version is an arithmetic progression, but A.V. Sigal also considers other methods of specifying it [45]);

different variations of the ABC method (e.g., the share of a resource in the total quantitative or cost volume of purchases).

These weights, as noted above, need to be regularly reviewed.

It is worth noting that the author's proposed discrepancy coefficient IDi can also be used to assess the quality of supply chain management. For this purpose, integral discrepancy coefficient IN is introduced:

(9)

As with all the indicators proposed above,

However, condition IN = 0 describes a situation in which there are no deviations between actual deliveries and current needs, whereas when IN = 1, there is a deviation between the values of actual deliveries and current needs across the entire range of deliveries. This is also true for the discrepancy coefficient JN, formula (2). The integral discrepancy coefficient does not provide information on the magnitude of these deviations. It only indicates their existence and their relative share in the overall range of deliveries.

Considering that

where P — the set of resource types for which there is a deviation between actual supplies and current needs, then formula (9) can be written in the form:

(10)

Clearly, the proposed methodology can be used to assess the conformance of not only actual supplies with current needs, but also actual supplies with planned ones. In this case, instead of value Di of current needs for the i-th resource, value Oi of planned supplies for the i-th resource should be used.

Assessing the rhythm of resource supply and product output. The uniformity of the fulfillment of the supply plan for resources of the i-th type (or the production plan for the i-th type of product) is typically assessed using the rhythmicity index Ri. The basic formula for determining it is as follows:

(11)

where m — number of periods for which the rhythmicity index is determined; Rij — rhythmicity index of supplies (production) of the i-th resource (product) in the j-th period:

(12)

where Fij — actual volume of deliveries (output) of the i-th type of resource (product) during the j-th period; PSi — planned volume of deliveries (output) of the i-th type of material resources (products) during the j-th period.

When Ri = 1, there is complete conformance between actual deliveries and planned deliveries, while when Ri = 0, there is complete nonconformance.

Formula (12) can be rewritten as follows:

(13)

The numerator of formula (13) is obviously similar to formula (4), but the denominator is specified differently.

When calculating the rhythmicity index, only those types of material resources whose deliveries are included in the plan should be taken into account, that is, those for which the condition i ∈ F is satisfied. Let G = n(F), then the integral rhythmicity index R for all G types of material resources is calculated using the formula:

(14)

The methodology for assessing the rhythm of production and supply based on formulas (11)–(14) has the following shortcomings:

according to this methodology, an excess of actual supplies (or actual production volume) over planned ones does not lead to a decrease in the rhythmicity index (i.e., it does not have a negative impact on the rhythm and planned nature of production processes), although in reality this is not the case. Above-plan supplies impose additional strain on enterprises' production and transport-logistics infrastructure, raise costs, and create risks of resource and finished product losses (due to, e.g., storage difficulties and the inability to use them in a timely manner for their intended purposes), while also undermining the quality of interaction among value chain participants. These negative effects, according to the author, should be taken into account when assessing the rhythm of production. The scientific literature contains attempts to account for excess production (so‑called 'plan overload') [3]. However, when using the methods proposed in these studies, the value of the rhythmicity index can exceed 1, which does not correspond to the basic limitations described above;

when calculating the rhythmicity index, the deliveries (output) of those types of resources (products) whose purchases (production) are not provided for in the plan are not taken into account. Formula (12) does not allow these types of resources (products) to be included in the calculation of the rhythmicity index since the planned volumes for them are equal to zero. Nevertheless, the presence of unplanned output affects the rhythmicity index, reducing its value [46]. At the same time, the output of products not included in the plan is a relatively typical situation, caused by the need to adapt the production plan to actual demand (as is the supply of resources not originally provided for in the plan).

Thus, we can talk about two factors that affect the value of the rhythmicity index:

deviation of the actual output of planned product types from the planned values (in the methodology presented in formulas (11)–(14), only negative deviations are taken into account);

output of unplanned products (not taken into account).

It should be noted that there is an alternative approach to determining the supply rhythmicity index, which allows considering the negative impact of excess supplies on rhythm [6]. In accordance with this approach, the alternative rhythmicity index RAij for the i-th type of resource during the j-th period is calculated using the following formula:

(15)

The alternative rhythmicity index RAi for m periods is determined by formula (11), and for all types of material resources — by formula (14) with the replacement in these formulas of the standard rhythmicity indexes Rij and Ri by the alternative rhythmicity indexes RAij and RAi, respectively.

However, although the alternative rhythmicity index RAij takes into account the negative impact of excess deliveries on rhythm, as shown by formula (15), for Fij > 2PSij it takes on negative values, which lacks obvious management meaning. It is implied that formula (15) can be transformed to the form:

(16)

but, unlike formula (12), the condition Fij > 2PSij also does not have a clear management meaning, and its theoretical justification is difficult.

To assess the timeliness of deliveries (production) of the i-th type of material resources (products) during the j-th period, the author proposes using the modified rhythmicity index MRij:

(17)

Formula (17) shows that, in contrast to the standard supply rhythmicity index [3][6], the proposed modified rhythmicity index takes into account the fact that the excess of actual deliveries over planned ones has a negative impact on rhythm. Moreover, from formula (17), it follows that a larger volume of actual deliveries, compared to planned ones, has a lesser negative impact than a failure to fulfill the supply plan of comparable absolute value (Table 1). This asymmetry between the impact of underdeliveries and overdeliveries on supply rhythm was not anticipated when developing the modified rhythmicity index MRij. However, it accurately reflects the real-world specific features of production and logistics requirements. Although overdeliveries negatively impact logistics processes (due to the need, as noted above, to divert additional resources to service excess material flows), they may be associated with compensation for underdeliveries from previous periods. Furthermore, additional supply volumes (or production, since the rhythmicity index is also used to assess the consistency of product output) may be required to accelerate plan fulfillment or to build up reserves for future periods. Conversely, underdelivery (or plan underfulfillment) can result in production downtime and breach of customer obligations, potentially leading to significant financial losses. Due to this, underfulfillment of the plan can be considered to have a greater negative impact on the rhythm of production than overfulfillment, which is reflected in formula (17). This, in the author’s opinion, can serve as additional confirmation of the conformance of the proposed methodology for assessing the rhythm of production with the requirements of industrial enterprises.

Formula (17) is a further development of formulas (12) and (13), in which not only the numerator of formula (13), but also its denominator is reduced to a form similar to formula (4).

In addition, formula (17) allows calculating the modified rhythmicity index for those actually produced (delivered) types of products (resources), whose release (purchase) was not provided for in the plan. The value of the modified rhythmicity index is equal to zero in accordance with formula (17) due to the equality of the planned production volume (order) to zero.

In other words, formula (17) eliminates the uncertainty (division by 0) that arises in formula (12) for products (resources) not included in the plan. It is clear that either only the planned output (order) or only the actual production (purchases) volume can be equal to zero. Items formally represented in the enterprise's product range, but not included in the plan and not actually produced, are not taken into account when calculating the modified rhythmicity index (this also applies to purchased resources). In other words, the rhythmicity index, in accordance with formula (17), is determined for actual output (purchase volume) and for the compiled plan, and not for the full possible range of products (deliveries) of the enterprise.

It is clear that

(18)

In this case, when MRi = 1, all deliveries (production output) are carried out on time, in accordance with the plan, whereas when MRi = 0, the delivery (production) plan is not met. Thus, the modified rhythmicity index has a transparent management meaning – in contrast to the approach presented by formula (16).

Moreover, no artificial restrictions are imposed on the value of the modified rhythmicity index, unlike formulas (12) and (16). In other words, condition (18) is satisfied in accordance with the calculation method, and not due to deliberately imposed restrictions on the indicator values. This suggests that it is free from the methodological shortcomings inherent in traditional approaches to calculating the rhythmicity index.

The values of the rhythmicity indexes calculated by various methods are presented in Table 1.

To calculate the modified rhythmicity index MRl for m periods for the l-th type of product, the following formula was used – by analogy with formula (11):

(19)

The integrated modified rhythmicity index IMR for all types of resources will be calculated by analogy with formula (14) using the following formula:

(20)

where L — total number of types of products actually produced by the enterprise during m periods:

where bj — number of types of products not included in the production plan, but actually produced during the j-th period.

Formula (20) can be used both in the basic version and to assess the rhythm of production only for those types of products whose production is planned (in accordance with the traditional approach to assessing the rhythm of production). In this case, the number of types of products for which the rhythm of production is assessed in formula (20) should be taken equal to G, not L.

Since L ≥ G, the author's proposed methodology allows for the negative impact of unscheduled production on production rhythm unlike formula (14). This can be confirmed using a hypothetical example (Table 2).

Based on the data in Table 2, three options of the rhythmicity index were calculated (Table 3):

the traditional rhythmicity index, formula (14), for two product types;

the modified rhythmicity index for product types scheduled for production, formula (20), items 1 and 2;

the modified rhythmicity index for all manufactured product types, formula (20), items 1, 2, 3, 4.

As shown in Table 3, the proposed methodology allows for the negative impact of unplanned output on production flow to be taken into account. In this example, the negative impact of unplanned output exceeds significantly the negative impact of deviations in actual output from planned values. This is explained by the fact that the terms corresponding to unplanned production in formula (20) are always equal to 0, while they lead to an increase in the number of types of products L, that is, to a decrease in the factor 1/L. Thus, the more types of unplanned products are produced, the lower the integral modified rhythmicity index.

Taking into account formulas (11), (17), (19) and (20), the expanded formula for calculating the value of the integral modified rhythmicity index can be written in the following form:

(21)

Formula (21) generalizes the stages of determining the value of the integral modified rhythmicity index.

Relationship between the proposed indicators and binary similarity measures for descriptive sets. It is important to emphasize that formula (7) is not a straightforward arithmetic mean of the partial conformance indicators ICi, formula (4), but rather an alternative approach to calculating the Jaccard index. If we consider the expression for calculating the Jaccard index for descriptive sets, formula (3), that is, the Ruzicka coefficient, and normalize each term in its numerator and denominator by max (Fi, Di):

(22)

then it is easy to verify that formula (22) is equivalent to formula (7). Thus, the integral similarity index TIC for N resource types is constructed by analogy with the Jaccard index calculated for normalized values of supplies of individual types of resources. Obviously, a similar argument can be applied to the modified rhythmicity index MRl, formula (19).

Formula (22) shows the relationship between the proposed indicators and classical binary measures of set similarity. Formula (22) itself represents another approach to calculating the Jaccard index, along with its variants for Boolean sets (Tanimoto coefficient [47]), sets with interval values of elements [48], etc. In other words, the conformance indicators and the modified rhythmicity index are part of the family of Jaccard indexes. This version of the Jaccard index can be called the Jaccard index for descriptive sets with normalized element values (for brevity, although this is not entirely correct from a mathematical point of view, it can also be called the normalized Jaccard index). It should be used to assess the similarity of those types of descriptive sets for which qualitative differences between their elements are important (in particular, differences in units of measurement). In other words, the specific nature of the normalized Jaccard index is determined not by the mathematical procedure for its calculation, but by the characteristics of the descriptive sets for which it is used to compare (the presence of qualitative differences between the elements), which determines the selection of calculation method. The author proposes to call such descriptive sets immiscible (due to the impossibility of simple summation of the elements included in them), in contrast to the traditionally considered descriptive sets for which summation of elements is permissible (such sets can be called miscible).

In most binary measures of similarity of descriptive sets, of which a significant number have been proposed to date [37][39], the presence of qualitative differences between elements is not taken into account, due to which, when assessing similarity, the values of elements are summed up (in other words, we are talking exclusively about mixed sets). On the contrary, in the proposed method, not an integral (the sum of the minimum and maximum values of the elements), but a component-by-component (for each pair of elements of the compared sets) assessment of similarity is performed. This makes it possible to take into account the qualitative differences of the elements (i.e., the specific features of immiscible sets).

These qualitative differences in the elements should not be confused with the qualitative differences in the scales in which the values of the elements are determined (nominal, ordinal, etc.). These differences in scales are taken into account in various versions of the Gower coefficient, in which all scales are reduced to one [37][49]. However, after this, the comparison of sets is performed using a method similar to the calculation of the Jaccard index, that is, by summing the values of the elements reduced to a single scale.

It should be noted that the normalization of the values of the elements is also performed when calculating the Gower coefficient, but the normalization procedure differs significantly from that proposed by the author (for the Gower coefficient, the difference between the maximum and minimum values of the elements is used), and it is used exclusively to reduce qualitatively different scales to a scale of the same type [37][49].

According to the author, this allows concluding that the proposed version of the Jaccard index, the normalized Jaccard index (22), represents a new measure of similarity of descriptive sets with qualitatively different elements. Moreover, the connection between the proposed indicators (the supply conformance indicator and the modified rhythmicity index) and binary measures of set similarity can be considered as a theoretical basis for the developed methods for assessing the production and logistics activities of an industrial enterprise.

Discussion. The integral conformance indicator TIC (7) and the integral modified rhythmicity index IMR (20) when used to assess the performance of the system of departments responsible for the material and technical support of the enterprise (i.e., for organizing purchasing activities and incoming logistics), actually represent an assessment of the provision of the company's departments with material resources according to the criteria of nomenclature and timeliness of their delivery. Furthermore, these indicators can also be used to evaluate the production activity of the enterprise, since they are easily adapted to determine the degree to which the nomenclature and composition of actually manufactured products corresponds to planned values and to assess the rhythm of production. For the modified rhythmicity index IMR, this is shown directly in the description of its calculation methodology. Finally, they can also be used to assess the degree to which actual deliveries made by a company correspond to customer demands (which can be both external customers and subsequent stages of a vertically integrated company's production chain), that is, to determine the performance of a company's outbound logistics. In the second case, we are talking about assessing the quality of interactions between various departments within a vertically integrated structure.

Thus, the potential for using the set of indicators proposed by the author extends beyond assessing the quality of work of the enterprise's MTS departments and can be applied at various stages of production. Company management can standardize these indicators.

In addition to the proposed key performance indicators for assessing the quality of an industrial enterprise's logistics and production activities (integrated conformance indicator TIC and integrated modified rhythmicity index IMR), the author also developed additional indicators. The feasibility of using the Jaccard index J to assess the conformance of deliveries according to the supply nomenclature (1) was demonstrated, the integrated nonconformance coefficient IN (7) was introduced, etc. These indicators can also be used to evaluate a company's performance. This allows the management of the enterprise (or its individual divisions) to select the optimal indicator based on current objectives.

A summary set of indicators developed by the author is presented in Table 4.

A key challenge is developing measures to improve these indicators (i.e., to improve the company's production and logistics activities). According to the author, one of the key tools for addressing this challenge in the context of the transition to a digital technological paradigm is digital transformation of the enterprise based on the use of the Internet of Things, big data, predictive analytics, and artificial intelligence [50]. These technologies make it possible to monitor the resource needs of enterprise departments in real time, predict the future level of needs and carry out flexible planning of production and logistics activities [21] (including taking into account seasonality [51]). However, it should be noted that, according to experts, the production infrastructure of Russian industry is currently characterized by a low level of readiness for the introduction of digital technologies [52].

From the above and in accordance with the recommendations presented in [35], an important conclusion follows that the indicators proposed by the author can be used to evaluate the efficiency of measures for the digital transformation of an industrial enterprise (or the implementation of digital technologies in its individual divisions). For this purpose, it is possible to calculate the data on the relative increase in the integral conformance indicator TICG and the integral modified rhythmicity index IMRG:

where the lower coefficient 0 corresponds to the value of the indicator before the implementation of digital transformation activities, and the lower coefficient 1 corresponds to the value of the indicator after these activities.

Obviously, the following condition should be satisfied:

(23)

Condition (23) is convenient in that it characterizes the goals of digital transformation of an industrial enterprise.

An important consequence (beyond the scope of this article) is that the proposed method for calculating the normalized Jaccard index (22) represents another approach to determining the value of this coefficient (along with existing methods for calculating the Jaccard index for sets with elements specified in a nominal scale [39], Boolean sets [53], data with interval uncertainty [48], probabilistic assessment [54], etc.) and allows evaluating the similarity of descriptive sets for which the presence of qualitative differences between elements is of fundamental importance.

Finally, another consequence, also beyond the scope of this article, but of undoubted interest, is the division (as far as is known, not previously presented in the literature) of descriptive sets into miscible and immiscible. Existing binary measures of similarity of descriptive sets were proposed exclusively for miscible sets. The normalized Jaccard index presented by the author allows evaluating the similarity of immiscible sets. Obviously, by analogy (in accordance with the methodology of B.I. Semkin [39]), analogs of other similarity measures developed for miscible sets (Sørensen coefficient, etc.) can be introduced for immiscible sets.

Conclusion. The research conducted by the author allows for the formulation of the following conclusions:

when evaluating the performance of an enterprise's MTS department, it is insufficient to consider only actual material resource deliveries. Account should also be taken of the full range of both supplies and needs — including those resource types that the enterprise needs but which are not included in the supply plan;

the author has proposed an integrated indicator of the conformance of actual supplies to current needs, which makes it possible to solve an important problem that had not previously been posed in general, namely, to determine the degree of conformance of the actual supplies of material resources (according to their nomenclature and the number of individual types of material resources) performed by the company's MTS department in a certain period, to the current needs of the enterprise (or its division) in the same period;

a methodology has been developed for calculating the modified rhythmicity index, which is free from the shortcomings characteristic of existing approaches to determining the rhythmicity index, since it takes into account the negative impact of above-plan deliveries on rhythm, and there are no artificial restrictions on the value of the indicator;

it has been shown that the implementation of digital technologies can improve the values of the author's proposed assessment of industrial enterprises' production and logistics activities, as they contribute to improved supply chain planning and greater flexibility in supply chain management. These indicators can also be used to justify the feasibility of digital transformation and to assess the magnitude of its impact.

From a theoretical point of view, the scientific novelty of this research lies in the fact that to develop a methodology for calculating indicators of production and logistics activities of an enterprise, the apparatus of set theory is used. Planned and actual supply volumes and planned and actual output volumes for all types of products are considered as descriptive sets. This allows moving from intuitive and empirical methods for assessing the rhythm of production to a theoretically based approach. At the same time, the author has proposed an alternative version of the Jaccard index (the normalized Jaccard index), which provides evaluating the similarity of those descriptive sets for which the presence of qualitative differences between elements is of fundamental importance (the term “descriptive immiscible sets” can be used to designate such sets).