where:

*WQ*—machinery operational efficiency indicator,

*Qr*—actual (achieved) efficiency (pcs./h),

*QO*—theoretical efficiency, as defined in the technical documentation (pcs./h).

Thus calculated, it indicates the degree of efficient use of the production line for each operation and, as such, indicates areas for improvement. Efficiency is most often presented in percentage form. This does not always allow for its quick and unambiguous assessment. In forecasting studies, it is assumed to be a quantitative variable, which limits the availability of some modeling methods. Therefore, with regard to the analyzed company, according to the authors, a better approach would be to analyze the efficiency from the individual point of view of each company by determining its satisfactory level and reacting only if it is not achieved.

Numerous studies using logistic regression models with regard to machine maintenance are available in the literature. The main objective of the proposed tools is to assess the technical condition of technical objects along with reliability parameters [30,31], predict upcoming failures [32,33] and estimate the service life of machinery [34]. For example, Yan and Lee assessed the performance of an elevator door system in real time and identified the types of possible failures [30]. Kozłowski et al. developed a model classifying the condition of a cutting tool blade and predicting its durability [31]. Lee et al. studied the reliability of a cutting tool using a combination of logistic regression and acoustic emission methods [32]. Caesarendra combined methods of logistic regression and relevance vector machine to evaluate performance degradation and to predict failure times based on simulation and experimental data [33], whereas Chen et al., on the basis of vibration characteristics of cutting tools, developed a universal model enabling the analysis of reliability and performance for machine tools [34].

This article proposes a model of logistic regression to be used for analysis and evaluation of the level of efficiency of executed processes. The research covered the process of manufacturing garbage bags in a company operating several production plants located in Poland and Ukraine. It was carried out in three main stages, in line with the CBM (condition-based maintenance) strategy. The basis for the research were the work and inspection cards of roll making machines provided by the company, which came from one of the plants and covered the period from 1 September 2015 to 31 August 2017. These provided information in two main categories. Event data indicated what events occurred during the operation of the machine (i.e., the need to repair, replacement of worn parts or breakdowns). On the other hand, the condition monitoring data provided information about the current technical condition of the facilities and the need for preventive measures (e.g., adjustment of Teflon blades). The processing of the above information and interpretation thereof made it possible to identify factors shaping the efficiency of the machinery stock. Then, on their basis, a model for the evaluation of machinery efficiency was built. It was assumed that its satisfactory level was 90%. This value is based on the daily production cycle, which also includes the breaks required by the Labor Code, daily service and the preparation of the machine for operation. Finally, the manner in which the model can support decision making in the area of improvement of the production processes was indicated [35].

Due to the specificity of manufactured products, i.e., serial products with standard parameters, the company operates in the MTS (make to stock) production system. The plant works on a three-shift basis, with each shift lasting 8 h. The process of model parameter estimation and the results obtained are presented in subsequent sections of this article.

#### **2. Logistic Regression Model**

Logistic regression is a model that allows the influence of several variables *X*1, *X*2, ... , *Xk* on the dichotomous variable *Y* in the mathematical form to be presented. The logistic regression model is based on a logistical function that takes the following form:

$$f(\mathbf{x}) = \frac{e^{\mathbf{x}}}{1 + e^{\mathbf{x}}} = \frac{1}{1 + e^{-\mathbf{x}}},\tag{2}$$

where *e* is the Euler number, and *x* is the value of the explanatory variable *X*.

The use of logistic regression is supported by the fact that it is not required to meet many assumptions that are formulated in relation to linear regression and general linear models. These include, first of all, the linearity of the relationship between a dependent and an independent variable, as well as the normality and homoscedasticity of the distribution of independent variables. In addition, observations must be reported using metric measurement systems.

The logistic regression model can be written in several ways. Assuming that *Y* stands for a dichotomous variable with values 1, for the occurrence of the event we are interested in (success), and 0, for the opposite case (failure), the logistic regression model is described by Equation (3):

$$P(Y = 1 | \mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_k) = \frac{e^{\beta\_0 + \sum\_{i=1}^k \beta\_i \cdot \mathbf{x}\_i}}{1 + e^{\beta\_0 + \sum\_{i=1}^k \beta\_i \cdot \mathbf{x}\_i}},\tag{3}$$

where β*<sup>i</sup> i* = 0, ... , *k* are logistic regression factors, while *x*1, *x*2, ... , *xk* are independent variables, which can be measurable or qualitative.

An equivalent form of the logistic regression equation can be written as the odds for the occurrence of the event (success) we are interested in:

$$\frac{P(Y=1|X)}{1 - P(Y=1|X)} = e^{\beta\_0 + \sum\_{i=1}^{k} \beta\_i \cdot x\_i}.\tag{4}$$

In a special case, for one independent variable the logistic regression equation takes the following form:

$$P(Y=1|X) = \frac{e^{\beta\_0 + \beta\_1 \cdot x\_1}}{1 + e^{\beta\_0 + \beta\_1 \cdot x\_1}}.\tag{5}$$

If, in turn, both sides of the Equation (5) are logarithmized, the logit form of the logistic model will be obtained:

$$\text{logit}\,P(Y=1|X) = \ln \frac{P(Y=1|X)}{1 - P(Y=1|X)} = \beta\_0 + \beta\_1 \cdot x\_1. \tag{6}$$

The condition necessary for logistic regression is a sufficiently large sample, the number of which should be *n* > 10(*k* + 1), where *k* is the number of parameters.

Important concepts related to logistic regression are the odds and the odds ratio. The odds are defined as the probability of an event occurring *P*(*A*) divided by the probability of an event not occurring, 1 − *P*(*A*):

$$P(\text{Odds})S(A) = \frac{P(A)}{P(non-A)} = \frac{P(A)}{1 - P(A)}.\tag{7}$$

The odds ratio, in turn, marked *OR*, is defined as the odds of one event occurring *S*(*A*) divided by the odds of another event occurring *S*(*B*):

$$OR\_{\text{AxB}} = \frac{S(A)}{S(B)} = \frac{P(A)}{1 - P(A)} : \frac{P(B)}{1 - P(B)}.\tag{8}$$
