Multi-Criteria Optimization of the Paper Production Process Using Numerical Taxonomy Methods: A Necessary Condition for Predicting Heat and Electricity Output in a Combined Heat and Power (CHP) System

Abstract

The subject of this study is the optimization of the paper production process in one of Poland’s leading paper mills. In addition to its primary objective of paper production, the company generates heat and electricity for internal consumption and external clients, including the local municipality. Surplus energy may be sold on the power exchange; however, this requires forecasting the quantity of energy to be sold 24 h in advance, which introduces an element of uncertainty. Production stoppages, often caused by random events such as paper breakage, force a power decrease in the CHP system, further complicating energy forecasting. To minimize the occurrence of such events, numerical taxonomy methods were employed to determine the optimal screen speed (V_s) and winding speed (V_n) for two paper machines, based on the type and weight of the paper produced. This analysis utilized detailed daily data collected by the company over the period 2015–2020. The findings contribute to minimizing the occurrence of paper roll tearing, thereby reducing the risk of inaccurate forecasts of the energy and heat produced by the CHP system. Furthermore, the methodology employed in this study may be effectively applied to other optimization problems in industrial processes.

1. Introduction

The term “taxonomy” is derived from the Greek word “taksis” (meaning ordering) and “nomos” (meaning principle). In the English-language literature, the term “taxonomy” or “numerical taxonomy” are commonly employed, while in American academic discourse, the term “cluster analysis” is more prevalent. The term “cluster analysis” is also found in the Polish literature [1,2].

Taxonomic methods are applied across various scientific fields. In medicine, for example, they support the classification of viruses and bacteria based on clinical characteristics [3,4] and have been used to organize healthcare guidelines during the COVID-19 pandemic [5,6,7]. In the natural sciences, these methods assist in modeling and visualizing phenomena such as cloud formation [8].

In the economic and social sciences [9], the topics addressed can vary widely, yet the methods employed remain rooted in taxonomy. This approach encompasses various analyses and classifications. For instance, one can analyze capital markets, economic strategies, spatial analyses, and quality of life [10]. In Slovenia, researchers group municipalities into socio-economically homogeneous clusters, revealing developmental differences between the Eastern and Western regions of the country [11]. Similarly, in Portugal, continental regions are classified by socio-economic indicators to inform regional development policies [12].

Furthermore, a multidimensional approach is used to assess regional disparities in Greece, uncovering uncovered the limited development convergence that occurred from 1995 to 2007 [13]. Global cluster policies are also investigated, with a focus on the socio-economic and institutional factors that influence regional clusters, as illustrated by a case study on Russia [14]. In the Volyn region of Ukraine, rural regions are classified based on their socio-economic conditions, employing spatial clustering to evaluate the development potential of farms [15]. Additionally, taxonomic methods are applied to analyze socio-economic factors in both social and economic research [16], and multidimensional analysis is utilized to assess and categorize socio-economic development levels across different regions [17].

Despite these diverse topics, these studies demonstrate a common methodological foundation in taxonomic analysis: to classify and understand socio-economic patterns across various contexts.

The authors of these studies focused on identifying and analyzing regional economic inequalities, employing cluster analysis to group territorial units according to different socio-economic indicators. These analyses helped to identify the factors that drive regional development and understand the differences between regions:

‑

In capital market analysis, strategy building, space analysis, and quality-of-life analysis [9,18];

‑

In the analysis of investment portfolios [19,20,21].

In investment portfolio analysis, taxonomic methods have been applied to classify assets for improved diversification. For example, one study clusters companies from the LQ-45 index to form optimized portfolios by grouping stocks with similar characteristics [19]. Another study employs an iterative algorithm to create a “fundamental portfolio” with minimized risk, using semivariance to enhance diversification [20]. Additionally, clustering methods applied to the Russell 1000^® index demonstrate that clustered portfolios achieve greater stability and lower volatility compared to Mean-Variance Optimization [21].

Asset diversification has been demonstrated to be an effective strategy for reducing risk, and in certain cases, taxonomic methods have proven more efficient than traditional approaches [22,23]. Moreover, taxonomic methods have shown effectiveness as tools for product measurement and optimization [24].

In the realm of technical and engineering sciences, research has focused on the optimization of the enrichment process for different lithological types of Polish copper ore [25,26]. This study examined the impact of a four-dimensional vector of independent variables on a three-dimensional vector of flotation process outcomes, ultimately leading to process optimization through numerical taxonomy. Additionally, taxonomic methods have been combined with deep learning approaches for wind and solar energy forecasting [27]. In the energy sector, a chapter in a monograph utilized cluster analysis to conduct a statistical comparison of Poland’s energy security with other European Union countries [28]. Another study applied hierarchical methods based on the Euclidean metric and taxonomic grouping to evaluate European countries in terms of energy transition [29].

In military sciences, researchers have introduced a new approach to analyzing complex multivariate objects using numerical taxonomy. This method is illustrated through the example of a “master object”, which serves as a reference point for comparing other objects [30].

The research presented here was carried out on the basis of historical data collected for the years 2015–2020 from a large paper plant in Poland, which is also a combined heat and power plant producing heat and electricity not only for its own needs but also for external customers. In this plant, the necessity to manage the generation processes occurrs in a way that allows for the sale of surplus energy on the energy market. A key requirement for such sales is the accurate prediction of heat and electricity generation.

However, under the plant’s operational conditions, this process is hindered by unexpected disruptions in paper production, which result in a reduction in available heat in the gas–steam system. These disruptions arise from various factors, including sudden irregularities in the paper roll, which necessitate the release of steam by the paper machine, leading to periods of machine downtime. Consequently, minimizing these operational interruptions would substantially improve the accuracy of energy production forecasting at any given time. Such optimization can be accomplished by adjusting the paper production process based on the type and weight of the paper, as well as the machine’s primary operational parameters, specifically:

• Screen speed V_s;
• Winding speed V_n.

The objective of this study is to determine, for a given type and weight of paper, the optimal values of V_s and V_n that would minimize machine downtime, thereby enhancing the efficiency and continuity of the paper production process. Numerical taxonomy methods were applied to achieve this optimization.

2. Materials and Methods

In the paper mill under study, due to the significant energy demands of the paper production process, heat and electricity are generated using a gas–steam cogeneration system. This system offers several advantages, including increased efficiency of energy generation, reduced costs of energy production, and a corresponding decrease in the cost of purchasing emission allowances and minimized environmental impact [31].

In the CHP system, the desired operational parameters are achieved by configuring vents and reduction-cooling stations within the steam section.

Priority for heat utilization is given to the MP1 and MP2 paper machines, which operate continuously and require a steady supply of heat 24 h a day.

2.1. Paper Mill

The selected paper mill is one of the largest producers of offset paper in Poland. It specializes in the production of modern paper designed for high-speed inkjet printing technology, as well as uncoated, wood-free paper that is extensively used in various printing applications, such as the production of brochures, envelopes, books, and documents. The company employs over 400 individuals and continuously strives to improve cost efficiency, emphasizing the efficient use of natural resources and minimizing environmental footprint.

In addition to paper production, the plant’s economic activities encompass the generation of heat and electricity, as well as the selling of surplus energy to meet the demands of both internal and external customers. Heat, in the form of steam, is supplied to various end-users, including the following:

• Pulping process;
• MP1 paper machine;
• MP2 paper machine;
• External companies;
• The plant’s own internal energy requirements;
• Exchangers, which distribute hot water for heating purposes to a range of facilities, such as the following:

  ‑

  The paper-processing department;

  ‑

  The product and raw material preparation department;

  ‑

  Paper machine exchangers;

  ‑

  Finished product warehouse;

  ‑

  Office buildings;

  ‑

  On-site stores;

  ‑

  The municipal heating network.

A challenge faced by the company is the instability of the system caused by sudden “jerks” in the paper machines, which occur when a roll of paper breaks. This results in an interruption to the machine’s operation, leading to the discharge of steam, which forces the machine to halt. To maintain the functioning of the steam turbines or to minimize their power output, steam must be diverted to specially designed blowout mechanisms. These sudden disruptions in the paper machines complicate the prediction of useful heat and electricity generated by the system. Therefore, minimizing the frequency of these “jerks” and the associated machine downtime is essential for improving system efficiency.

2.2. Numerical Taxonomy

Numerical taxonomy is based on a multidimensional comparative analysis, which involves the classification and ordering of multifeature statistical objects, each described by a specific set of variables (features) [30,32]. The application of the taxonomic method can be divided into several stages:

• Stage I—Formation of the Diagnostic Feature Matrix

The first step involves constructing a matrix of diagnostic features, denoted as C:

$$\mathrm{C}=\left[\begin{array}{cc}\begin{array}{ccc}{\mathrm{C}}_{11}& {\mathrm{C}}_{12}& \dots \\ {\mathrm{C}}_{21}& {\mathrm{C}}_{22}& \dots \\ \begin{array}{c}\dots \\ {\mathrm{C}}_{\mathrm{k}1}\end{array}& \begin{array}{c}\dots \\ {\mathrm{C}}_{\mathrm{k}2}\end{array}& \begin{array}{c}\dots \\ \dots \end{array}\end{array}& \begin{array}{c}{\mathrm{C}}_{1\mathrm{n}}\\ \begin{array}{c}{\mathrm{C}}_{2\mathrm{n}}\\ \begin{array}{c}\dots \\ {\mathrm{C}}_{\mathrm{k}\mathrm{n}}\end{array}\end{array}\end{array}\end{array}\right]$$

(1)

where ${\mathrm{C}}_{\mathrm{i}\left(·\right)}=\left[{\mathrm{C}}_{\mathrm{i}1},{\mathrm{C}}_{\mathrm{i}2},\dots ,{\mathrm{C}}_{\mathrm{i}\mathrm{n}}\right]$—feature vector of the i-th object in k objects, $i\in \left\{1,\dots ,\mathrm{k}\right\}$; ${\mathrm{C}}_{\left(·\right)\mathrm{j}}=\left[{\mathrm{C}}_{1\mathrm{j}},{\mathrm{C}}_{2\mathrm{j}},\dots ,{\mathrm{C}}_{\mathrm{k}\mathrm{j}}\right]$—vector of the j-th feature, $\mathrm{j}\in \left\{1,\dots ,\mathrm{n}\right\}$.

• Stage II—Normalization of Features

Next, to standardize the numerical values of features and normalize (or standardize) the vectors of individual features, the feature vectors (i.e., columns of matrix C) are normalized. Two common methods are as follows:

‑

Normalization [25,26]:

$${\mathrm{x}}_{\mathrm{i}\mathrm{j}}={\displaystyle \frac{{\mathrm{C}}_{\mathrm{i}\mathrm{j}}}{\underset{\mathrm{i}}{\max }\left({\mathrm{C}}_{\mathrm{i}\mathrm{j}}\right)}},\mathrm{i}\in \left\{1,\dots ,\mathrm{k}\right\}\mathrm{j}\in \left\{1,\dots ,\mathrm{n}\right\}$$

(2)

‑

Standardization [30]:

$${\mathrm{x}}_{\mathrm{i}\mathrm{j}}={\displaystyle \frac{{\mathrm{C}}_{\mathrm{i}\mathrm{j}}-\overline{{\mathrm{C}}_{\mathrm{j}}}}{{\mathsf{\sigma }}_{\mathrm{j}}}},\mathrm{i}\in \left\{1,\dots ,\mathrm{k}\right\},\mathrm{j}\in \left\{1,\dots ,\mathrm{n}\right\}$$

(3)

where

$$\overline{{\mathrm{C}}_{\mathrm{j}}}={\displaystyle \frac{1}{\mathrm{k}}}\sum _{\mathrm{i}=1}^{\mathrm{k}}{\mathrm{C}}_{\mathrm{i}\mathrm{j}}$$

(4)

$${\mathsf{\sigma }}_{\mathrm{j}}=\sqrt{{\displaystyle \frac{1}{\mathrm{k}}}\sum _{\mathrm{j}=1}^{\mathrm{k}}{\left({\mathrm{C}}_{\mathrm{i}\mathrm{j}}-\overline{{\mathrm{C}}_{\mathrm{j}}}\right)}^2}$$

(5)

This process results in a new matrix $\mathrm{X}={\left({\mathrm{x}}_{\mathrm{i}\mathrm{j}}\right)}_{\mathrm{k}\mathrm{x}\mathrm{n}}$.

• Stage III—Determination of the Pattern Vector

For the matrix X, the coordinates of the pattern vector w are determined:

$$\mathrm{w}=\left[{\mathrm{w}}_1,\dots ,{\mathrm{w}}_{\mathrm{n}}\right]$$

(6)

The pattern vector coordinates can be determined, for example, as follows [30,33,34]:

For stimulants (features where higher values are better):

$${\mathrm{w}}_{\mathrm{j}}=\underset{\mathrm{i}}{\max }\left({\mathrm{x}}_{\mathrm{i}\mathrm{j}}\right)\mathrm{for}\mathrm{j}\in \left\{1,\dots ,\mathrm{n}\right\}$$

For destimulants (features where lower values are better):

$${\mathrm{w}}_{\mathrm{j}}=\underset{\mathrm{i}}{\min }{\mathrm{x}}_{\mathrm{i}\mathrm{j}}\mathrm{for}\mathrm{j}\in \left\{1,\dots ,\mathrm{n}\right\}$$

In cases where the features are standardized, the following is assumed:

$${\mathrm{w}}_{\mathrm{j}}=1,\mathrm{j}\in \left\{1,\dots ,\mathrm{n}\right\}$$

(7)

• Stage IV—Calculation of Distances

The distances of $\mathrm{d}\left({\mathrm{x}}_{\mathrm{i}\left(·\right)},\mathrm{w}\right)$ for each row of the matrix X (representing the studied objects) from the pattern vector w are calculated sequentially. The most commonly used distance metric is the Euclidean distance, though other distance metrics such as the Manhattan distance (formerly referred to as “taxi distance”), maximum distance, Mahalanobis distance, or the hyperbolic distance can also be employed [2,35,36,37]. By comparing these distances, the objects can be ranked in either ascending or descending order.

For stimulants, the optimal value is the largest value (i.e., the smallest distance from the pattern), whereas for destimulants, the optimal value is the smallest (i.e., the greatest distance from the pattern) [38].

2.3. Adopted Methodology

In this study, taxonomic methods based on the Euclidean metric were utilized for a multi-criteria analysis and classification of data based on their quantitative characteristics.

These methods were applied to optimize the smoothness of the paper production process, which has a direct impact on the accuracy of predicting the amount of heat and electricity produced by the plant. Taxonomic methods using Euclidean metrics facilitate the hierarchical ordering of the studied objects.

3. Experiment

In this analysis, for the given ranges of screen speed (V_s) and winding speed (V_n) during the year under review, for a specific type of paper, the following total times were recorded [2]:

• Production time (T1);
• Downtime (T2);
• Assortment change time (T3);
• Spurt time (i.e., machine jerking or stopping) (T4).

Subsequently, the probabilities of these events were calculated as follows:

• Production probability:

  $${\mathrm{P}}_1={\displaystyle \frac{{\mathrm{T}}_1}{\sum {\mathrm{T}}_{\mathrm{i}}}}$$

  (8)
• Downtime probability:

  $${\mathrm{P}}_2={\displaystyle \frac{{\mathrm{T}}_2}{\sum {\mathrm{T}}_{\mathrm{i}}}}$$

  (9)
• Assortment change probability:

  $${\mathrm{P}}_3={\displaystyle \frac{{\mathrm{T}}_3}{\sum {\mathrm{T}}_{\mathrm{i}}}}$$

  (10)
• Spurt probability:

  $${\mathrm{P}}_4={\displaystyle \frac{{\mathrm{T}}_4}{\sum {\mathrm{T}}_{\mathrm{i}}}}$$

  (11)

The goal was to find such configurations of V_s and V_n that would simultaneously yield ${\mathrm{P}}_{1\left(\mathrm{m}\mathrm{a}\mathrm{x}\right)}$ and ${\mathrm{P}}_{2\left(\mathrm{m}\mathrm{i}\mathrm{n}\right)}$, ${\mathrm{P}}_{3\left(\mathrm{m}\mathrm{i}\mathrm{n}\right)}$, and ${\mathrm{P}}_{4\left(\mathrm{m}\mathrm{i}\mathrm{n}\right)}$.

Since P₁ is a stimulant (higher values are better), and P₂, P₃, and P₄ are destimulants (lower values are better), feature $\mathrm{Q}={\displaystyle \frac{1}{{\mathrm{P}}_1}}$ was introduced to transform P₁ into a destimulant [39]. In this way, the task became one of minimizing Q, P₂, P₃, and P₄.

In the first stage, based on historical data collected from 2015 to 2020, the full ranges of variability in V_s and V_n for each specific paper type produced each year were determined. These ranges were divided into six smaller classes (intervals) to create a matrix of probabilities for each event within the given speed ranges.

The next step involved normalizing the data (for Q, P₂, P₃, and P₄) using the formula

$$\mathrm{v}=\left({\displaystyle \frac{\mathrm{Q}}{{\mathrm{Q}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}},{\displaystyle \frac{{\mathrm{P}}_2}{{\mathrm{P}}_{2\left(\mathrm{m}\mathrm{a}\mathrm{x}\right)}}},{\displaystyle \frac{{\mathrm{P}}_3}{{\mathrm{P}}_{3\left(\mathrm{m}\mathrm{a}\mathrm{x}\right)}}},{\displaystyle \frac{{\mathrm{P}}_4}{{\mathrm{P}}_{4\left(\mathrm{m}\mathrm{a}\mathrm{x}\right)}}}\right)$$

(12)

The reference vector is $w=\left(1,1,1,1\right)$.

Finally, the Euclidean distance $\mathrm{D}=\mathrm{d}\left(\mathrm{v},\mathrm{w}\right)$ was calculated as follows:

$$\mathrm{D}=\mathrm{d}\left(\mathrm{v},\mathrm{w}\right)=\sqrt{{\left(1-{\displaystyle \frac{\mathrm{Q}}{{\mathrm{Q}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}}\right)}^2+{\left(1-{\displaystyle \frac{{\mathrm{P}}_2}{{{\mathrm{P}}_2}_{\mathrm{m}\mathrm{a}\mathrm{x}}}}\right)}^2+{\left(1-{\displaystyle \frac{{\mathrm{P}}_3}{{{\mathrm{P}}_3}_{\mathrm{m}\mathrm{a}\mathrm{x}}}}\right)}^2+{\left(1-{\displaystyle \frac{{\mathrm{P}}_4}{{{\mathrm{P}}_4}_{\mathrm{m}\mathrm{a}\mathrm{x}}}}\right)}^2}$$

(13)

for each year, for each paper type, and for various ranges of V_s and V_n.

This study analyzed data collected from 2015 to 2020, covering various paper types: A-60, A-70, A-80, A-90, A-100, B-70, B-756, B-80, C-70, C-80, and C-90.

Table 1. Probability matrix for A-80 paper in 2020 [2].

A-80, Year 2020
Vs [m/s] Vn [m/s]	I [835–854]	II [855–874]	III [875–893]	IV [894–912]	V [913–931]	VI [932–950]
880–898 (1)	P1 = 0.9324
	P2 = 0.0389
	P3 = 0.0056
	P4 = 0.0231
899–917 (2)		P1 = 0.8404
		P2 = 0.0998
		P3 = 0.0064
		P4 = 0.0534
918–936 (3)		P1 = 0.0882	P1 = 0.5795
		P2 = 0.9108	P2 = 0.3948
		P3 = 0.001	P3 = 0.0046
		P4 = 0	P4 = 0.0211
937–955 (4)				P1 = 0.7465
				P2 = 0.2224
				P3 = 0.0035
				P4 = 0.0276
956–974 (5)				P1 = 0.8689	P1 = 0.8891
				P2 = 0.0863	P2 = 0.0697
				P3 = 0.0074	P3 = 0.0033
				P4 = 0.0374	P4 = 0.0379
975–992 (6)					P1 = 0.8667	P1 = 0.8978
					P2 = 0.0913	P2 = 0.0836
					P3 = 0.0036	P3 = 0.0087
					P4 = 0.0384	P4 = 0.0099

presents the results of subsequent stages of calculations for the production of A-80 paper for 2020. A summary of the best results for subsequent years, paper types, and both paper machines (MP1 and MP2) is provided for comparison.

This analysis focuses on the probability matrix, which was developed for specific ranges of sieve speed (V_s) and winding speed (V_n) to evaluate the occurrence of particular phenomena during the paper production process. It was determined that only certain velocity ranges—those where specific types of production phenomena were observed—could be considered for analysis.

Table 2. Comparison of the considered velocity ranges of V_s and V_n with the values of the probabilities of the considered phenomena and the calculated distance for A-80 paper in 2020.

A-80 2020
Object	Class Vs	Class Vn	P1	$\mathit{Q}=\frac{1}{{\mathit{P}}_1}$	P2	P3	P4	D
O1	And	(1)	0.932	1.073	0.039	0.006	0.023	1.478205
O2	II	(2)	0.84	1.19	0.1	0.006	0.053	1.289908
O3	II	(3)	0.088	11.34	0.911	0.001	0	1.335413
O4	III	(3)	0.58	1.726	0.395	0.005	0.021	1.275807
O5	IV	(4)	0.747	1.34	0.222	0.004	0.028	1.392694
O6	IV	(5)	0.869	1.151	0.086	0.007	0.037	1.318661
O7	V	(5)	0.889	1.125	0.07	0.003	0.038	1.460736
O8	V	(6)	0.867	1.154	0.091	0.004	0.038	1.427912
O9	VI	(6)	0.898	1.114	0.084	0.009	0.01	1.517104

provides a comparison of the considered ranges for sieve speed (V_s) and winding speed (V_n) with the corresponding values of the probabilities of the key phenomena (production, downtime, assortment change, and spurt) and the Euclidean distance calculated for each range. This is exemplified using A-80 paper from the year 2020.

Based on the calculated distances, the objects can be ranked in either ascending or descending order. In descending order, the ranking is as follows: O9, O1, O7, O8, O5, O3, O6, O2, and O4. It is visualized in Figure 1.

By applying this method, a ranking of objects—specifically the velocity ranges of V_s and V_n—was obtained. The optimal configuration corresponds to the range with the greatest normalized distance from the reference vector. The comparative results are summarized in

Table 3. Comparison of V_s and V_n based on the Euclidean distance criterion (ranked from best to worst).

Class		Mid-Class
Vs [m/s]	Vn [m/s]	Vs [m/s]avg	Vn [m/s]avg
932–950	975–992	941	983.5
835–854	880–898	844.5	889
913–931	956–974	922	965
913–931	975–992	922	983.5
894–912	937–955	903	946
855–874	918–936	864.5	927
894–912	956–974	903	965
855–874	899–917	864.5	908
875–893	918–936	884	927

.

The aforementioned ranking demonstrates that the optimal results for P₁(max), P₂(min), P₃(min), and P₄(min) were achieved for object O9, corresponding to the range of 932–950 for V_s and 975–992 for V_n.

Similar calculations were performed for each type of paper considered across the years 2015 to 2020 for both paper machines. Using the same methodology, the optimal velocity ranges of V_s and V_n were determined for each case. The results are summarized in

Table 4. List of the best results for the MP1 paper machine [2].

	2015		2016		2017		2018		2019		2020		Average
	Vs	Vn	Vs	Vn	Vs	Vn	Vs	Vn	Vs	Vn	Vs	Vn	Vs	Vn
A-60	778.5	838	755.5	810.5	781	824	766	827	889	934.5	860.5	929	805.1	860.5
A-70	815	869	819	869	789.5	860.5	755	801	831.5	856.5	931.5	981	823.6	872.8
A-80	793	841.5	805.5	875	756	819	767.5	828	819.5	886.5	941	983.5	813.8	872.3
A-90	745.5	797	740.5	802	756	781.5			772.5	820	952.5	992.5	793.4	838.6
A-100	705	746.5	688	728							891	922	761.3	798.8
B-70	789	835			799.5	854.5	782	832	890.5	915	883	924.5	828.8	872.2
B-75			803	846	787.5	842.5	788	846.5	810.5	872.5	899	946	817.6	870.7
B-80					764.5	821.5	800	844.5	870.5	917.5	899.5	934	833.6	879.4
C-70	764	816	825.5	870	779	828	808	858	839	878			803.1	850.0
C-80	783	811.5	780.5	823	763.5	813.5	759	810	871.5	910.5	888	953	807.6	853.6
C-90	766.5	814.5	786.5	807	711	747	765.5	816	841	881.5	861.5	903	788.7	828.2

and

Table 5. List of the best results for the MP2 paper machine [2].

	2015		2016		2017		2018		2019		2020		Average
	Vs	Vn	Vs	Vn	Vs	Vn	Vs	Vn	Vs	Vn	Vs	Vn	Vs	Vn
A-80	754	801.5	778	813.5	776.5	813	717.5	759.5	783.5	836	741.5	812	758.5	805.9
A-90	810	851	793.5	851	759.5	816.5	751.5	768.5	801.5	834.5	744	758.5	776.7	813.3
A-100	718.5	771	709	738.5	690	745.5	688	752	653.5	716	645	710.5	684.0	738.9
A-110	713	771.5	692.5	719	720.5	776.5	680	744.5	665.5	724	625.5	694.5	682.8	738.3
A-120	645.5	703	659.5	663	663	716	652	687	681	745	574	635.5	645.8	691.6
A-130	592.5	639.5	617	662.5	634.5	684	644	682.5	583	617	513	548	597.3	638.9
A-140	530.5	552.5	540.5	582.5	545	563.5	579.5	607	548	575.5	577	604	553.4	580.8
A-150	470	514	476.5	482.5	468	480.5	497	498.5	487.5	509	449	474	474.7	493.1
A-170	476	488	475.5	507	475.5	499.5	475.5	486	419	455	462.5	489.5	464.0	487.5
B-70	790.5	837.5	794.5	836	788.5	826	800	837	768.5	823	791	842	788.8	833.6
B-75	810.5	838.5	799	851	781.5	813	791	822.5	754	788.5	789	842	787.5	825.9
B- 90	745.5	776.5	740.5	796.5	764.5	814.5	757.5	804.5	727	778	736.5	785	745.3	792.5
B-100	724.5	744.5	748	782	758	800.5	770	791.5	718	756.5	718	748	739.4	770.5
B-120			585	622.5	597	605	621.5	643	649	690.5	602	625	610.9	637.2
B-150			454.5	481	454.5	492	463	492	444	472.5	453	486	453.8	484.7
B-160	483.5	503	468	477	464.5	477.5	483.5	504.5	468	509.5	449	485.5	469.4	492.8
D-80			755.5	793	781	835.5	688	718.5	747.5	780	680.5	738	730.5	773.0
D-90			738	777.5	688	746.5	635	692.5	653.5	686.5	710.5	749.5	685.0	730.5
E-80			784	833.5	717.5	752.5	719.5	754.5	744	800	700.5	737.5	733.1	775.6
E-90							670	701	616	645	679.5	717.5	655.2	687.8

(Table 4 pertains to the MP1 paper machine, while Table 5 pertains to the MP2 paper machine). The symbols A, B, C, D, and E represent specific types of paper, and the numerical values that follow (60, 70, 80, etc.) denote the weight of the paper.

Based on the data presented in Table 4 and Table 5, trend equations were established to illustrate the variation in velocity V_s and V_n in relation to the type of paper for the years under consideration.

The graphs depicting these relationships for the MP2 paper machine, covering the years 2015 to 2020 for paper type A, are collectively summarized in Figure 2.

Analogous graphs were created for other types of paper machines and are included in Appendix A (Figure A1 and Figure A2). These visual representations facilitate a clearer understanding of how velocity adjustments correlate with the specific paper types produced during the examined period.

The graphs presented here illustrate a trend whereby the values of the optimal velocity ranges of V_s and V_n exhibit a decreasing pattern as the weight of the paper increases. The resulting R² values for all regression equations are notably high, approximately 0.9, indicating a strong correlation between paper grammage and the optimal speed ranges. This suggests that as the grammage of a given type of paper increases, the corresponding values of the optimal speed ranges of V_s and V_n decline.

Furthermore, for each paper type (A, B, C, D, and E), the relationship between the screen and winding speed values (V_s and V_n) was examined for each year considered. These relationships are characterized by a linear course and a very high R² value, confirming that these speeds are closely interrelated.

Figure 3 depicts the regression lines representing V_s as a function of V_n for type “A” paper over the years 2015 to 2020. Additional diagrams illustrating similar relationships for other paper types are included in Appendix A (Figure A3 and Figure A4).

The developed regression equations facilitate the determination of screen speed values based on winding speed values. These findings corroborate previously observed trends.

The presented analyses demonstrate that as the grammage of a given type of paper increases, the optimal values of the speed ranges of V_s and V_n decrease. Consequently, it is possible to select rigid velocities V_s and V_n in a manner that mitigates the risk of undesirable phenomena, such as spurts. Such spurts significantly affect the quality and continuity of production, and addressing them serves as both the starting point and a preliminary step toward a more accurate prediction of the amount of useful heat and electricity produced in the cogeneration process.

4. Conclusions

The application of the numerical taxonomy method utilizing Euclidean distance yielded the following outcomes:

• Creation of a ranking of the ranges of screen speed V_s and winding speed V_n;
• Determination of the optimal ranges of V_s and V_n, which maximize production probability while minimizing the probabilities of interruptions due to machine breaks, spurts, changes in assortment, and downtime.

The analysis of the optimal screen speeds (V_s) and winding speeds (V_n) obtained for individual types of paper and their grammage revealed the following:

• As the grammage increases (for a given type of paper), the optimal values of the speed ranges of V_s and V_n decrease;
• The optimal velocities V_s and V_n are strongly correlated;
• The established regression equations enable the determination of screen speed based on a known winding speed value, thereby minimizing the risk of undesirable phenomena.

The identified optimal ranges of screen speeds V_s and winding speeds V_n ensure the following:

• Minimization of the risk of undesirable phenomena (such as spurts);
• Significant improvements in production quality and continuity;
• The potential for developing a more accurate prediction of the amount of useful heat and electricity generated.

The authors intend to continue their research on the issues addressed in this study.