Application of the Multi-Criteria Method FUCOM for Evaluating Technological Processes

Abstract

In modern industrial production, the selection and evaluation of technological processes is a factor in achieving high quality, efficiency, and sustainability. Due to the existence of numerous and often contradictory criteria, the decision-making process requires the application of reliable multi-criteria methods. This article demonstrates the application of MCDM (Multi-Criteria Decision-Making) methods, the FUCOM (Full Consistency Method), for evaluating and selecting a rational technological process under real production conditions. The research results presented in the article demonstrate that the FUCOM method ensures a high degree of consistency, transparency, and efficiency in the evaluation of technological processes. It allows, among a variety of alternative technological process for manufacturing a given product, for the clear identification of the most rational one according to specified requirements. The data obtained in a real production environment confirm the applicability of the method in the field of production engineering and provide a basis for future research and optimization of technological processes.

1. Introduction

Today, mechanical engineering production is characterized by a high degree of complexity and a variety of technological processes. The management of technological processes increasingly depends on assessing and choosing logical solutions using several, frequently incompatible criteria in light of the growing demands for efficiency, sustainability, and competitiveness. The finding of efficient technological solutions is greatly aided by contemporary Computer-Aided Manufacturing (CAM) systems, which integrate automated planning and production optimization techniques [1]. Every day, manufacturing firms must make decisions about technologies, equipment, materials, and organizational solutions that successfully balance technical, financial, and environmental factors. In such situations, multi-criteria decision-making (MCDM) methods can be used, which offer a systematic and reliable approach to evaluating and selecting optimal solutions [2,3,4,5,6,7,8].

Multi-criteria methods are analytical techniques that allow for the simultaneous evaluation of several alternatives according to multiple criteria, which are often contradictory—for example, low price, high quality, etc. Determining the weights of the criteria and processing the expert assessments are an essential stage in multi-criteria analysis, as the reliability, objectivity, and consistency of the final decision depend on them.

The most widely used MCDM methods [9] include AHP (Analytic Hierarchy Process), ANP (Analytic Network Process), TOPSIS (Technique for Order Preference by Similarity to Ideal Solution), VIKOR (Višekriterijumsko kompromisno rangiranje), SAW (Simple Additive Weighting), BWM (Best Worst Method), and the relatively newer FUCOM (Full Consistency Method) [3,10,11,12,13,14,15,16,17,18], as shown in Figure 1.

Each method has specific advantages and disadvantages, meaning that careful selection is required for their application in different areas. For example, AHP is a well-established and understandable approach, but it requires a large number of comparisons and is sensitive to subjectivity [19,20]. ANP takes into account the interrelationships between criteria, but is extremely complex to apply [21,22]. TOPSIS and VIKOR are good for use in situations with many quantitative criteria, but assume independence between them [16,23,24]. SAW and BWM are easy to use but are limited in complex dependencies [25,26,27,28,29]. Against this background, FUCOM stands out for its efficiency in working with a large number of criteria and low computational complexity [30,31].

These methods are used in various fields—from strategic management and resource planning to logistics, transport, energy, and risk assessment. Despite their advantages, they are often associated with high computational complexity, substantial demands on experts in terms of the number of comparisons, as well as the potential for inconsistencies in the assessments [32,33].

A comparative analysis of MCDM methods shows that each of them has different advantages and limitations depending on the specific objective and conditions of application [9,34]. The AHP method offers a clear hierarchical structure and is easy to understand, but there is a lack of subjectivity in determining the weights. ANP, as its enhancement, allows for the modeling of interdependencies between criteria, but requires significant resources and time for calculations [35]. TOPSIS [36] and VIKOR [34] are distinguished by their effectiveness in analyzing alternatives through their proximity to the “ideal solution,” which makes them suitable for engineering applications with quantitative data. SAW and BWM [37] offer lower computational complexity and are preferred for quick assessments and a small number of criteria. When choosing a method, it is important to consider the complexity of the problem, the available resources, and the degree of accuracy required.

The FUCOM method approach has become more well-known and popular in recent years due to its many benefits [38,39]. Its primary goal is to ensure complete consistency in the assigned weights while reducing the number of expert comparisons required. This makes it particularly attractive for practical applications, as it reduces the cognitive load on experts, limits the possibility of errors, and shortens the time required to perform the evaluation. In addition, FUCOM is sufficiently flexible and can be integrated both independently and in combination with other multi-criteria analysis methods. The method’s weaknesses include the requirement of normalizing quantitative criteria and its reliance on expert evaluations (“weakness” here refers not to the need for expert assessment itself, but to the possibility that it may be carried out by specialists who are not well prepared and have little or irrelevant experience). This calls for meticulous planning and validation of outcomes. Nevertheless, FUCOM is an up-and-coming tool for further study and applications due to its high consistency and adaptability.

Despite the existing examples of FUCOM application in various fields—logistics, transportation, supply chain management, strategic planning, and evaluation of alternatives [40,41,42]—there is a lack of research articles, papers, and practical research focused on its use for evaluating technological processes. This gap in the published research highlights the need for new advances in this field, which has drawn interest. Complex structures, many interdependencies, and particular evaluation criteria—typically including sustainability, energy efficiency, economic efficiency, and environmental indicators—are characteristics of technological processes. This makes them a particularly suitable area for the application of FUCOM, which can provide a reliable, transparent, and systematic mechanism for setting priorities and evaluating alternatives.

The main objective of this article is to propose and test the application of the FUCOM method for evaluating technological processes in real production conditions. This study aims to prove its applicability in a new direction, identify its possible advantages and limitations, and provide practical guidelines for future use in an industrial environment. Additional emphasis is placed on enriching the scientific literature with results that will serve as a basis for further research and development of multi-criteria methods for evaluating technological processes.

2. Materials and Methods

2.1. Methodology for Selecting a Rational Technological Process

To achieve the set goals, a methodology for selecting a rational technological process [43], already developed by the authors, is used. This is based on the application of the FUCOM method and served as the basis for conducting the present experimental study.

Figure 2 shows the sequence of the five main stages of the methodology for selecting a rational technological process. They are interrelated, logically arranged, and contribute to the optimization of production decisions and the achievement of high quality at minimum cost and time.

2.1.1. Stage 1—Development of Different Technological Processes

This stage aims to create several possible technological solutions (compliant with the objectives and hard requirements specified in the drawings and technical specifications) for manufacturing the same part. Their development takes into account production resources, equipment capabilities, production volume, and product quality requirements. Having more alternatives creates conditions for more flexible and rational decision-making.

$$P=\left\{P_1,P_2,P_3\dots P_m\right\},$$

(1)

where

-

m—the number of technological processes analyzed.

2.1.2. Stage 2—Define the Evaluation Criteria

In the second stage, the main criteria are defined, according to which the technological processes developed in stage 1 will be compared. They cover both technical indicators (accuracy, roughness, dimensional stability) and economic and organizational factors (production costs, production time, productivity). A clear definition of the criteria is key to achieving an objective and repeatable assessment.

$$C=\left\{C_1,C_2,C_3,\dots {,C}_n\right\}$$

(2)

where

-

n—index indicating the number of criteria considered.

2.1.3. Stage 3—Determining the Weighting of Criteria for Each Technological Process

After identifying the criteria, their relative importance is determined by applying the FUCOM method. The weights of the criteria are determined once by FUCOM and are applied equally to all technological processes under consideration. The method does not generate separate weights for the different variants of technological processes but provides a uniform, mathematically consistent system of weight coefficients used for comparison between them, and the set of weights is the same for all the alternatives. This allows for a high degree of consistency in expert assessment and minimizes subjectivity in setting priorities. The end product is a collection of normalized weighting coefficients that show how important each criterion is in relation to the others.

$$C_k=K,K\in \left\{1,2,\dots n\right\},$$

(3)

where $C_k\ne C_p,\forall p\in \left\{\mathrm{1,2},\dots n\right\},p\ne k,k=\mathrm{1,2},\dots n$.

2.1.4. Stage 4—Evaluation, Comparison, and Selection of the Rational Technological Process

At this point, all technological processes created in Stage 1 are evaluated using a mathematical model based on the weights determined in Stage 3 and the criteria established in Stage 2. Based on the designated indicators, each process is assigned a numerical value indicating its efficiency. The process with the best overall rating is determined to be rational, as it provides an optimal balance between quality, time, and price.

$$V_{i,j}=(P_i,{\omega }_j)$$

(4)

where

-

I = 1…m, j = 1…n, ${\omega }_j$ is the normalized value of the criterion C_j.

$$S\left(P_{1\dots m}\right)=\sum _1^n{\omega }_j·V(P_{1\dots m},C_j)\to min,$$

(5)

where

-

ω_j—weight assigned to each criterion C_(1…n) (the FUCOM—determined weight of the j-th criterion).

In manufacturing environments and when evaluating technological processes, the aim is always that criteria such as price and production time are as low as possible. Accordingly, all other criteria are normalized using the PN index, which minimizes deviations in dimensions, roughness, and other factors to a single number representing the deviations in the technological process for the criterion under consideration (i.e., more deviations means the normalized value is higher). All criteria are equalized so that the lower value is better, which allows Equation (5) to be minimized (i.e., the technological process with the minimum value according to the equation is the most suitable).

2.1.5. Stage 5—Planning the Implementation of the Selected Rational Technological Process in Manufacturing and Monitoring It

The final stage covers the actual implementation of the selected technological process in a production environment, as well as monitoring its results. Tracking statistical indices of process capability (Cp, Cpk, Pp, Ppk) allows for the timely detection of deviations, an assessment of stability, and the planning of corrective actions. This ensures sustainability and continuous improvements in production.

The suggested methodology ensures transparency, objectivity, and adaptability to the changing conditions of the modern production environment by combining strategic planning with a quantitative decision-making approach. The transparency argument comes from the fact that the proposed methodology allows for the traceability of the entire process, while objectivity is linked to reduced expert involvement in the assessment of the processes. This provides technology experts with a valuable tool for process optimization and sustainable quality, with reduced production risk.

2.2. Methodology for Conducting the Experiment

The planning of an experiment, choice of research object, defining of the criteria for assessing technological processes, selection of measurement tools, processing of the results, and choice of a logical technological process are all covered by the current experimental methodology. The main idea is to achieve a systematic approach—from the formulation of the research objectives and planning of the experiment, through conducting the experiments and measurements, to the final analysis and evaluation of the results. Following the steps of the methodology makes it possible to evaluate the quality of the final part, determine the influence of the various technological processes, and draw objective conclusions about their effectiveness.

Figure 3 shows a diagram of a specific methodology for conducting an experiment related to the implementation of the study in real production conditions at CERATIZIT Bulgaria AD. However, this methodology is universal and can be applied to other research subjects in other manufacturing companies.

The ability to systematically conduct experimental studies is ensured by adhering to the methodology’s steps. This, in turn, makes it easier to compare various technological solutions and encourages well-informed choices when designing technological processes.

2.2.1. Selection of Research Subject and Criteria

To test the methodology for selecting a rational technological process based on the application of the FUCOM method, the experimental part was carried out in the real production conditions of CERATIZIT Bulgaria AG.

The challenges in selecting a suitable research object include the limitations of the available production technologies and the need to balance the complexity of the part with the possibilities for its production in the manufacturing conditions of CERATIZIT Bulgaria AG. All of the above issues were discussed and resolved through iterative consultations and the active participation of the company’s technologists. With their help, one part—“Guide for 3D printer”, shown in Figure 4—was selected as the research object in connection with the testing of the methodology for selecting a rational technological process.

The object of the experimental research is a cylindrical part that acts as a guide for the parallels of a 3D printer. Its main purpose is to ensure precise guidance and minimize wear or friction between the contact surfaces. The part has clearly defined geometric dimensions, accuracy, and roughness parameters, which are essential for reliable and precise operation in the kinematic diagram of the 3D printer.

During the preliminary planning of the experimental research and in close cooperation with the technologists from CERATIZIT Bulgaria AG, six basic criteria were defined for the evaluation of each technological process. These criteria reflect the most important aspects of the production and quality of the resulting part:

• Internal diameter accuracy Ø16^+0.018 mm;
• External diameter accuracy Ø24 ± 0.2 mm, according to ISO 2768—m [44];
• Length accuracy 10 ± 0.2 mm, according to ISO 2768—m [44];
• Roughness of the machined surface with a required value of Ra 0.8 μm;
• Production costs;
• Manufacturing time.

All technological processes were evaluated according to the same criteria in order to ensure a uniform basis for comparison and to avoid inconsistencies arising from different methods of analysis. This makes the results directly comparable and allows conclusions to be drawn about which technological process is rational for specific production needs. This not only provides a realistic assessment of the capabilities of each process but also facilitates the selection of the optimal solution depending on predefined production or customer priorities.

2.2.2. Production Parameters

As part of the experiment, four different technological processes were designed in collaboration with technologists from CERATIZIT Bulgaria AG, each covering a set of different operations, machines, tools, and materials, and ensuring the production of suitable parts. For each of the four technological variants, 50 parts were produced to provide a sufficient basis for analysis of the technological processes. The characteristics of each technological process are summarized in

Table 1. Characteristics of different technological processes.

Characteristics	TP N° 1	TP N° 2	TP N° 3	TP N° 4
Material	41Cr4 (40X) 1.7035	41Cr4 (40X) 1.7035	55NiCrMoV7 1.2714	55NiCrMoV7 1.2714
Manufacturing costs	9.20 €	8.56 €	8.47 €	12.15 €
Manufacturing time, h	0.7006	0.6697	0.4447	0.2122
Workpiece, mm	Ø25	Ø25	Ø25.8	Ø25.8
Production line	WA CNC	WA	STW	DW

.

The number of parts produced by each technological process (50 pcs.) was chosen to be enough to provide an overview of the distribution of errors. In this way, we can obtain objective information about the degree of correspondence between the different technological processes.

To ensure full traceability and repeatability of the experimental results, detailed technical information on the measuring instruments used, calibration procedures, and experimental setup schemes is included in Appendix A. Descriptions of the SRT 6210 profilometer Graiger (Shenzhen, China), the ETOPOO digital micrometer Yiwu Hot Electronic Co. (Zhejiang, China), and the Caliper Mitutoyo (Kanagawa, Japan) are provided, along with their basic metrological characteristics and operating conditions.

3. Results

3.1. Measurement Results

After the production of the parts according to the four technological processes under examination, the main geometric and quality characteristics were measured in accordance with the criteria set out in the methodology for conducting experimental research (see Section 2.2). The purpose of the measurement is to determine the extent to which each process meets the requirements for accuracy and roughness. The measurements were performed on 50 pieces for each technological process. Each part was measured in a controlled environment using calibrated measuring instruments (see Appendix A).

The obtained summary data for each technological process are presented in

Table 2. Results from measurements of the different technological processes.

N°	TP N° 1 CNC_53-WA CNC 15043050				TP N° 2 C11-WA 15043057				TP N° 3 C11_47-STW 15043254				TP N° 4 STAMA-DW 15043227
	Ra 0.8, μm	10, mm	Ø24, mm	Ø16, mm	Ra 0.8, μm	10, mm	Ø24, mm	Ø16, mm	Ra 0.8, μm	10, mm	Ø24, mm	Ø16, mm	Ra 0.8, μm	10, mm	Ø24, mm	Ø16, mm
1	0.732	10.14	23.86	16.005	0.777	10.04	24.08	16.001	0.8	9.96	23.98	16.000	0.732	10.08	23.96	16.010
2	0.729	10.04	24.00	16.001	0.794	10.02	24.02	16.005	0.759	9.96	24.10	16.002	0.783	10.06	23.98	16.012
3	0.759	10.02	24.00	16.005	0.727	10.00	24.02	16.002	0.75	9.98	24.08	16.003	0.469	10.04	24.10	16.007
4	0.8	10.02	24.02	16.006	0.668	10.02	24.08	16.003	0.777	9.96	23.96	16.007	0.571	10.02	23.90	16.012
5	0.712	10.00	24.00	16.001	0.691	10.04	24.06	16.009	0.771	10.18	23.92	16.008	0.765	10.04	23.96	16.013
6	0.759	10.04	24.00	16.004	0.636	10.02	24.08	16.006	0.777	9.92	24.04	16.006	0.6	10.06	24.00	16.012
7	0.759	10.02	24.02	16.008	0.753	10.06	24.08	16.002	0.718	10.08	23.98	16.003	0.706	10.04	23.98	16.018
8	0.738	10.00	24.02	16.013	0.724	10.04	24.08	16.008	0.724	9.92	24.08	16.008	0.606	10.02	24.02	16.018
9	0.709	9.96	24.00	16.008	0.665	9.92	23.92	16.008	0.782	9.94	24.00	16.001	0.53	10.00	24.04	16.017
10	0.759	10.02	24.02	16.003	0.724	10.04	24.08	16.005	0.8	10.10	23.94	16.000	0.483	10.02	24.04	16.004
11	0.732	10.02	24.02	16.005	0.729	10.02	24.02	16.002	0.771	9.96	23.98	16.0038	0.76	10.04	23.88	16.004
12	0.706	10.00	24.02	16.009	0.8	10.04	24.02	16.007	0.747	10.04	23.98	16.0036	0.804	10.00	24.00	16.003
13	0.782	10.04	24.05	16.006	0.732	10.02	24.04	16.005	0.8	9.98	24.08	16.004	0.771	10.08	24.00	16.007
14	0.75	10.02	24.00	16.001	0.718	10.04	24.02	16.003	0.721	10.08	24.02	16.003	0.759	10.06	23.98	16.004
15	0.8	10.04	24.02	16.011	0.715	10.06	24.08	16.004	0.718	9.92	24.14	16.001	0.771	10.04	23.96	16.004
16	0.729	10.06	24.02	16.005	0.645	10.08	24.08	16.000	0.744	9.98	23.98	16.0039	0.6	10.08	24.00	16.009
17	0.782	10.04	24.00	16.000	0.665	10.00	24.06	16.002	0.777	9.96	23.96	16.004	0.8	10.08	24.02	16.001
18	0.735	10.02	24.02	16.008	0.633	10.04	24.00	16.004	0.765	9.84	24.00	16.001	0.62	10.00	24.00	16.003
19	0.732	10.02	24.02	16.005	0.765	10.02	24.10	16.003	0.759	9.96	23.98	16.0038	0.8	10.06	24.04	16.004
20	0.765	10.06	24.04	16.004	0.735	10.06	24.02	16.002	0.735	9.9	23.96	16.005	0.806	10.08	24.02	16.008
21	0.732	10.10	24.000	16.002	0.721	10.02	24.08	16.002	0.732	9.98	23.96	16.002	0.652	10.10	24.00	16.002
22	0.735	10.06	24.02	16.002	0.794	10.00	24.02	16.004	0.729	9.92	24.08	16.001	0.78	10.08	23.98	16.007
23	0.788	10.02	24.00	16.007	0.592	10.02	24.00	16.002	0.777	9.88	24.02	16.004	0.63	10.06	24.00	16.003
24	0.729	10.12	24.00	16.001	0.636	10.04	24.08	16.005	0.729	10.02	23.98	16.0037	0.606	10.10	24.00	16.004
25	0.788	10.04	24.02	16.003	0.636	10.02	24.04	16.001	0.724	9.92	23.98	16.0039	0.753	10.08	24.00	16.003
26	0.794	10.04	24.02	16.004	0.636	10.06	24.02	16.000	0.753	10.04	23.96	16.004	0.777	10.06	23.90	16.004
27	0.709	10.04	24.00	16.011	0.8	10.04	24.02	16.003	0.753	9.98	24.10	16.006	0.76	10.04	24.02	16.003
28	0.777	10.02	24.00	16.003	0.744	10.02	24.06	16.003	0.715	9.92	24.00	16.004	0.788	10.04	23.98	16.01
29	0.721	10.02	24.02	16.007	0.709	10.06	24.02	16.005	0.782	10.02	24.10	16.011	0.729	10.08	24.02	16.002
30	0.788	10.00	24.02	16.003	0.753	10.02	24.02	16.005	0.712	9.90	24.08	16.009	0.753	10.10	23.90	16.003
31	0.765	10.02	24.00	16.006	0.794	10.04	24.02	16.004	0.75	10.08	24.06	16.001	0.6	10.08	24.06	16.005
32	0.765	10.02	24.02	16.003	0.8	10.02	24.08	16.005	0.732	9.92	24.06	16.005	0.718	10.06	23.98	16.004
33	0.794	10.04	24.02	16.004	0.656	9.86	23.96	16.000	0.765	9.96	24.08	16.003	0.703	10.06	24.00	16.007
34	0.735	10.04	24.02	16.001	0.674	10.02	24.08	16.003	0.8	9.98	23.98	16.010	0.7	10.00	23.98	16.007
35	0.727	10.02	24.06	16.006	0.759	10.06	24.00	16.008	0.759	10.00	24.06	16.005	0.779	10.06	23.96	16.010
36	0.771	10.04	24.00	16.012	0.753	10.04	24.14	16.008	0.794	9.92	24.10	16.010	0.783	10.08	24.00	16.004
37	0.753	10.08	24.02	16.007	0.738	10.03	24.04	16.002	0.782	10.16	23.96	16.004	0.706	10.12	23.98	16.002
38	0.759	10.02	24.00	16.009	0.709	10.02	24.02	16.008	0.753	9.88	23.98	16.005	0.554	10.08	24.02	16.008
39	0.782	10.02	24.02	16.008	0.732	10.02	24.04	16.009	0.788	10.02	24.04	16.003	0.823	9.98	23.96	16.000
40	0.709	10.00	24.02	16.003	0.589	10.00	23.98	16.003	0.753	9.92	23.96	16.000	0.794	10.06	24.00	16.003
41	0.771	10.04	24.10	16.004	0.668	10.02	24.02	16.007	0.765	9.96	23.96	16.002	0.642	10.08	24.00	16.004
42	0.727	10.04	24.02	16.014	0.639	10.06	23.98	16.002	0.753	10.14	23.94	16.002	0.8	10.06	24.00	16.008
43	0.709	10.04	24.00	16.009	0.771	10.04	24.10	16.002	0.732	9.92	23.96	16.006	0.741	10.06	23.98	16.003
44	0.8	9.86	24.02	16.006	0.8	10.08	24.00	16.001	0.694	9.94	23.96	16.001	0.794	9.94	24.00	16.003
45	0.782	10.04	24.02	16.008	0.744	10.06	24.02	16.009	0.697	9.96	24.02	16.002	0.642	10.08	23.90	16.004
46	0.744	10.02	24.06	16.004	0.771	10.02	24.00	16.013	0.753	9.90	23.86	16.000	0.694	10.08	24.02	16.006
47	0.715	10.02	23.98	16.009	0.686	9.98	23.98	16.013	0.765	9.98	24.06	16.012	0.645	10.06	24.04	16.004
48	0.765	10.04	23.88	16.002	0.794	9.90	24.08	16.000	0.75	9.98	24.02	16.009	0.765	10.14	24.00	16.006
49	0.729	9.92	24.00	16.006	0.656	10.00	24.02	16.014	0.771	10.06	23.96	16.006	0.674	10.06	23.96	16.007
50	0.721	10.10	24.00	16.010	0.697	10.02	24.00	16.007	0.753	9.96	24.02	16.009	0.545	10.08	24.00	16.003

. Average values, standard deviations, and dispersion limits are calculated for each parameter. This allows for comparison between the individual technological processes not only by average value but also by stability of performance.

3.2. Analysis of the Results

3.2.1. Statistical Data Analysis

The measurement of the parameters studied shows that all four technological processes, despite their differences, ensure compliance with the tolerances specified in the technical documentation and guarantee the production of suitable parts. Figure 5 and Figure 6 show control charts of the measured parameters of the parts manufactured using the four technological processes. The control charts facilitate the analysis and comparison of the results for each of the four technological variants. To track the results obtained, the warning limits of the parameter in review (discussed in Section 3.2.2) are shown in orange.

The analysis of four different technological processes provides a clear picture of their stability and ability to maintain critical parameters within the specified tolerances. All parameters considered—diameters Ø16 mm and Ø24 mm, size 10 mm, and surface roughness Ra 0.8 μm—remain within acceptable limits, which indicates that the technological processes have been correctly selected and that quality control methods have been effectively applied.

However, the results show some differences in the behavior of the individual processes. With a well-defined uniform distribution around the nominal values and little variation in dimensions, TP N° 1 is clearly the most reliable and repeatable. This characterizes it as a reference procedure that can be used as a foundation for comparison in subsequent research. TP N° 2 and TP N° 3 also demonstrate good quality, but they show isolated peaks and values close to or above the control limits.

The greatest variation in dimensions is observed in TP N° 4, especially for the parameters Ø16 mm and Ra 0.8 μm. Although the process remains within tolerances, the high amplitude of the fluctuations is a sign of potential instability.

3.2.2. Determining the Warning Limits

Warning limits are defined values within the acceptable tolerance range that serve as an indicator of potential instability or deviation in the process.

They do not indicate non-compliance but signal an increasing likelihood that the process will exceed the specified limits if preventive action is not taken. According to ISO 7870-2:2023 [45], warning limits are determined by the following formulas:

$$\sigma ={\displaystyle \frac{USL-LSL}{6}}$$

(6)

$$UWL={\displaystyle \frac{USL+LSL}{2}}+2\sigma$$

(7)

$$LWL={\displaystyle \frac{USL+LSL}{2}}-2\sigma$$

(8)

where

-

USL—upper specification limit;

-

LSL—lower specification limit;

-

UWL—upper warning limit;

-

LWL—lower warning limit.

The formulas were used to calculate the warning limits for the following measured dimensions: Ø 16 mm, Ø 24 mm, and 10 mm. Their values are shown in

Table 3. Calculation of warning limits.

Parameter	UWL	LWL	Standard Deviation—σ	2 σ
Ø 16, mm	16.015	16.003	0.003	0.006
Ø 24, mm	24.13	23.87	0.133	0.0667
10, mm	10.13	9.87
Ra 0.8 µm	0.78	0.6	0.045	0.09

.

When setting specification limits for a parameter such as roughness (in this case, Ra 0.8 µm), the standards do not mention a specific method for calculating the limits. According to the roughness standard “EN ISO 21920-3 [46]”, if no other “acceptable” criterion is specified, the so-called maximum limit value rule applies by default. That is, for this specific case, Ra = 0.8 µm is specified in the drawing of the parts, which means that the upper limit is 0.8 µm and for values above this, the product is considered non-compliant with the drawing. The lower warning limit is not mentioned in the standard because all values below the specified upper limit are considered acceptable.

No standard has been found for calculating warning limits when measuring roughness, so the standard “ISO 7870-2:2023 [45]” is applied. In this case, it is not possible to take 0.8 µm as the average value because it is the upper control limit, so the center line ($\overline{X}$ = 0.6 µm) is selected to ensure a sufficient margin from the upper “engineering limit” (control upper limit—0.80 µm) and to ensure stable surface quality without the risk of exceeding the specification under normal process fluctuations. The calculated values according to the standard are presented in Table 3.

3.2.3. Analysis of Statistical Data

For the purposes of the methodology, the values of all parameters, which in this case are measured for 50 machined parts (i.e., there are 50 measured values), must be reduced to a single summary value that represents the quality and stability of the technological process. For this purpose, a normalized index—P_N—is introduced, defined by the following formula:

$$P_N=\left(1-{\displaystyle \frac{N_w}{n}}\right).100\%,$$

(9)

where

-

n—number of measured values;

-

N_w—number of measured values falling within the range LWL ≤ x ≤ UWL;

-

LWL—lower warning limit;

-

UWL—upper warning limit.

By calculating the percentage of measurements that fall outside the specified warning interval, the method provides an easily interpretable metric that allows for quick comparison between different technological processes. If most of the results are centered around the average value, the process is considered stable, and a small percentage of deviations indicates better quality. This early warning of small but significant deviations provides an opportunity for operational control and timely intervention before the process leads to more serious problems. In this way, the normalized index—P_N—not only combines all the information into a single number but also assists in the rational selection of the technological process by allowing a comparison to be made and for the selection of the option whose measurements are closest to the target value.

The normalization via P_N converts the various measured parameters into a dimensionless value based on the percentage deviation from the warning limits. This eliminates differences in measurement units, ranges, and tolerances, allowing for a correct comparison between different criteria and different technological processes. The normalized values obtained in this way are fully comparable and can be used for objective evaluation using the FUCOM method.

Table 4. Results from measurements of different technological processes.

Technological Process	TP N° 1 CNC_53-WA CNC 15043050				TP N° 2 C11-WA 15043057				TP N° 3 C11_47-STW 15043254				TP N° 4 STAMA-DW 15043227
Measured parameters	Ra 0.8, µm	10, mm	Ø24, mm	Ø16, mm	Ra 0.8, µm	10, mm	Ø24, mm	Ø16, mm	Ra 0.8, µm	10, mm	Ø24, mm	Ø16, mm	Ra 0.8, µm	10, mm	Ø24, mm	Ø16, mm
PN	24	4	2	30	20	2	2	48	18	8	4	40	34	2	0	36

shows the normalized values of the measured parameters—roughness (Ra = 0.8 µm) and dimensional accuracy (Ø16 mm, Ø24 mm, 10 mm)—using the normalized index—P_N.

3.3. Application of the Methodology for Selecting a Rational Technological Process

3.3.1. Stage 1—Designing Various Technological Processes

The four technological processes, developed in cooperation with the technologists of CERATIZIT Bulgaria AG, were designed for the manufacture of a part—guide for 3D printer. For each of them, batches of 50 parts were produced. The measured values of these parts are presented in Table 2.

3.3.2. Stage 2—Defining the Criteria

The criteria that will help evaluate and compare the technological processes were again selected in cooperation with specialists from CERATIZIT Bulgaria AG in order to fully comply with the approval process in real production conditions and the decision-making process of the specialist technologists. Production costs and manufacturing time are presented in Table 1, and Table 2 shows the measured values of all parts from the four technological processes.

3.3.3. Stage 3—Defining the Importance of the Criteria

Table 5. Assigning importance to each criterion.

Criterion	Weight
C1 Production costs	1
C2 Accuracy of Ø16+0.018 mm	2
C3 Accuracy of Ø24 ± 0.2 mm	3
C4 Accuracy of length 10 ± 0.2 mm	4
C5 Production time, h	5
C6 Roughness Ra 0.8 μm	6

shows the importance of the selected criteria used to evaluate the technological processes. If the criteria change, the selected rational technological process may change and be replaced by another technological process that meets the new criteria. The importance of the criteria can be determined both by specialists and by customer needs or requirements. The importance of the criteria is assessed by assigning 1 to the most important criterion and 6 to the least important.

The presented example uses a simplified form of FUCOM, in which the expert technologist provides only the rank of the criteria. This allows for inaccurate or arbitrary numerical ratios to be avoided, while maintaining the full consistency of FUCOM through the weighting model. Therefore, simplification does not violate the methodology but increases reliability and applicability in a real production environment.

• Application of the multi-criteria method

The weights of the criteria are calculated via the following formulas using the FUCOM multi-criteria method. By solving linear equations with many unknowns, the weights for each criterion that will be included in the evaluation of the technological processes are calculated.

$\left|\begin{array}{c}{\phi }_1={\displaystyle \frac{C_2}{C_1}}=2={\displaystyle \frac{{\omega }_1}{{\omega }_2}}\\ {\phi }_2={\displaystyle \frac{C_3}{C_2}}=1.5={\displaystyle \frac{{\omega }_2}{{\omega }_3}}\\ {\phi }_3={\displaystyle \frac{C_4}{C_3}}=1.33={\displaystyle \frac{{\omega }_3}{{\omega }_4}}\\ {\phi }_4={\displaystyle \frac{C_5}{C_4}}=1.25={\displaystyle \frac{{\omega }_4}{{\omega }_5}}\\ {\phi }_5={\displaystyle \frac{C_6}{C_5}}=1.2={\displaystyle \frac{{\omega }_5}{{\omega }_6}}\end{array}\right.$	$\left|\begin{array}{c}{\displaystyle \frac{{\omega }_2}{{\omega }_3}}.{\displaystyle \frac{{\omega }_1}{{\omega }_2}}={\displaystyle \frac{{\omega }_1}{{\omega }_3}}=3\\ {\displaystyle \frac{{\omega }_3}{{\omega }_4}}.{\displaystyle \frac{{\omega }_2}{{\omega }_3}}={\displaystyle \frac{{\omega }_2}{{\omega }_4}}=1.995\\ {\displaystyle \frac{{\omega }_4}{{\omega }_5}}.{\displaystyle \frac{{\omega }_3}{{\omega }_4}}={\displaystyle \frac{{\omega }_3}{{\omega }_5}}=1.6625\\ {\displaystyle \frac{{\omega }_5}{{\omega }_6}}.{\displaystyle \frac{{\omega }_4}{{\omega }_5}}={\displaystyle \frac{{\omega }_4}{{\omega }_6}}=1.5\end{array}\right.$
$\left|\begin{array}{c}{\omega }_1=3{\omega }_3\\ {\omega }_2=1.995{\omega }_4\\ {\omega }_3=1.6625{\omega }_5\\ {\omega }_4=1.5{\omega }_6\end{array}\right.$	$\left|\begin{array}{c}{\omega }_1=2{\omega }_2\\ {\omega }_2=1.5{\omega }_3\\ {\omega }_3=1.33{\omega }_4\\ {\omega }_4=1.25{\omega }_5\\ {\omega }_5=1.2{\omega }_6\end{array}\right.$	⇒	$\begin{array}{c}{\omega }_1=0.408\\ {\omega }_2=0.204\\ {\omega }_3=0.136\\ {\omega }_4=0.102\\ {\omega }_5=0.082\\ {\omega }_6=0.068\end{array}$
	${\displaystyle \sum _{j=1}^n}{\omega }_j=1$

3.3.4. Stage 4—Calculation and Comparison of the Different Technological Processes

Table 6. Quantitative values of each criterion for evaluation for the specific technological process.

Quantitative Data	Production Costs	Ø16, mm	Ø24, mm	10, mm	Production Time, h	Ra 0.8 μm
TP N°1/CNC_53	9.20 €	30	4	4	0.7006	24
TP N° 2/C11	8.56 €	48	2	2	0.6697	20
TP N° 3/C11_47	8.47 €	40	4	8	0.4447	18
TP N° 4/STAMA	12.15 €	36	0	2	0.2122	34

presents the quantitative values of each technological process, which include production costs, a normalized index of measured values, and manufacturing time.

$$S\left(P_m\right)=\sum _{j=1}^n{\omega }_j·V(P_m,C_j)$$

$$TP1/S\left(P_1\right)=12.520$$

$$TP2/S\left(P_2\right)=15.181$$

$$TP3/S\left(P_3\right)=14.241$$

$$TP4/S\left(P_4\right)=14.840$$

The criteria specified in

Table 7. Combinations of the importance of the selected criteria and final assessment of the technological process.

Criteria	N°	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5	Criterion 6	Rational Technological Process
		Production Costs	Ø16	Ø24	10	Time for Production	Ra 0.8 μm
Importance of criteria	1.	6	4	5	1	2	3	1. TP 1/S(P1) − 9.054 2. TP 2/S(P2) − 9.318 3. TP 4/S(P4) − 9.985 4. TP 3/S(P3) − 10.790
	2.	5	3	1	4	2	6	1. TP 4/S(P4) − 8.450 2. TP 1/S(P1) − 8.649 3. TP 2/S(P2) − 9.747 4. TP 3/S(P3) − 9.898
	3.	2	6	3	4	5	1	1. TP 3/S(P3) − 13.193 2. TP 2/S(P2) − 13.706 3. TP 1/S(P1) − 14.724 4. TP 4/S(P4) − 19.028
	4.	2	1	3	4	6	5	1. TP 1/S(P1) − 17.082 2. TP 4/S(P4) − 20.167 3. TP 3/S(P3) − 20.915 4. TP 2/S(P2) − 23.493
	5.	6	5	4	3	2	1	1. TP 3/S(P3) − 12.776 2. TP 2/S(P2) − 13.277 3. TP 1/S(P1) − 13.966 4. TP 4/S(P4) − 17.958
	6.	1	2	3	4	5	6	1. TP 1/S(P1) − 12.520 2. TP 3/S(P3) − 14.242 3. TP 4/S(P4) − 14.840 4. TP 2/S(P2) − 15.181
	7.	3	4	5	1	2	6	1. TP 1/S(P1) − 8.048 2. TP 4/S(P4) − 8.499 3. TP 2/S(P2) − 8.539 4. TP 3/S(P3) − 10.141
	8.	4	5	2	1	6	3	1. TP 2/S(P2) − 8.783 2. TP 1/S(P1) − 9.150 3. TP 4/S(P4) − 9.635 4. TP 3/S(P3) − 10.690
	9.	2	5	1	6	3	4	1. TP 2/S(P2) − 8.750 2. TP 1/S(P1) − 8.776 3. TP 4/S(P4) − 9.053 4. TP 3/S(P3) − 9.068
	10.	5	6	2	1	3	4	1. TP 2/S(P2) − 7.321 2. TP 4/S(P4) − 7.755 3. TP 1/S(P1) − 7.785 4. TP 3/S(P3) − 9.391
	11.	3	4	2	6	5	1	1. TP 3/S(P3) − 13.978 2. TP 2/S(P2) − 14.825 3. TP 1/S(P1) − 15.254 4. TP 4/S(P4) − 19.357
	12.	3	4	2	1	6	5	1. TP 1/S(P1) − 8.769 2. TP 4/S(P4) − 8.933 3. TP 2/S(P2) − 8.965 4. TP 3/S(P3) − 10.815
	13.	4	6	5	3	2	1	1. TP 3/S(P3) − 12.438 2. TP 2/S(P2) − 12.874 3. TP 1/S(P1) − 13.789 4. TP 4/S(P4) − 17.882
	14.	6	3	2	5	1	4	1. TP 1/S(P1) − 8.585 2. TP 4/S(P4) − 9.444 3. TP 3/S(P3) − 9.506 4. TP 2/S(P2) − 9.999
	15.	5	4	6	1	3	2	1. TP 1/S(P1) − 10.710 2. TP 2/S(P2) − 10.722 3. TP 3/S(P3) − 12.044 4. TP 4/S(P4) − 12.449
	16.	3	4	2	5	1	4	1. TP 4/S(P4) − 5.576 2. TP 1/S(P1) − 5.742 3. TP 3/S(P3) − 6.885 4. TP 2/S(P2) − 6.907
	17.	1	2	5	3	6	4	1. TP 1/S(P1) − 13.245 2. TP 3/S(P3) − 14.902 3. TP 2/S(P2) − 15.812 4. TP 4/S(P4) − 16.062
	18.	5	3	4	2	6	1	1. TP 3/S(P3) − 15.552 2. TP 1/S(P1) − 15.901 3. TP 2/S(P2) − 16.050 4. TP 4/S(P4) − 20.190
	19.	2	4	3	5	1	6	1. TP 1/S(P1) − 7.728 2. TP 3/S(P3) − 8.413 3. TP 2/S(P2) − 8.714 4. TP 4/S(P4) − 8.716
	20.	4	5	6	1	2	3	1. TP 2/S(P2) − 8.602 2. TP 1/S(P1) − 8.701 3. TP 4/S(P4) − 9.664 4. TP 3/S(P3) − 10.207
	21.	5	2	4	1	3	6	1. TP 1/S(P1) − 10.642 2. TP 4/S(P4) − 11.497 3. TP 2/S(P2) − 12.967 4. TP 3/S(P3) − 13.813
	22.	3	1	6	5	2	4	1. TP 1/S(P1) − 16.687 2. TP 4/S(P4) − 20.023 3. TP 3/S(P3) − 20.332 4. TP 2/S(P2) − 23.233
	23.	3	2	4	5	6	1	1. TP 3/S(P3) − 17.754 2. TP 1/S(P1) − 17.952 3. TP 2/S(P2) − 19.537 4. TP 4/S(P4) − 23.055
	24.	2	3	6	4	5	1	1. TP 3/S(P3) − 15.642 2. TP 1/S(P1) − 16.493 3. TP 2/S(P2) − 16.836 4. TP 4/S(P4) − 21.477
	25.	1	6	3	5	4	2	1. TP 3/S(P3) − 11.094 2. TP 2/S(P2) − 11.345 3. TP 1/S(P1) − 11.636 4. TP 4/S(P4) − 14.532
	26.	4	5	1	2	3	6	1. TP 4/S(P4) − 6.929 2. TP 2/S(P2) − 7.468 3. TP 1/S(P1) − 7.565 4. TP 3/S(P3) − 8.680
	27.	2	1	4	5	6	3	1. TP 1/S(P1) − 18.170 2. TP 3/S(P3) − 21.596 3. TP 4/S(P4) − 21.977 4. TP 2/S(P2) − 24.473
	28.	5	1	4	3	6	2	1. TP 1/S(P1) − 18.894 2. TP 3/S(P3) − 22.218 3. TP 4/S(P4) − 22.911 4. TP 2/S(P2) − 24.894
	29.	6	4	2	3	5	1	1. TP 3/S(P3) − 13.946 2. TP 2/S(P2) − 14.378 3. TP 1/S(P1) − 14.901 4. TP 4/S(P4) − 18.667
	30.	3	6	5	2	4	1	1. TP 3/S(P3) − 13.225 2. TP 2/S(P2) − 13.233 3. TP 1/S(P1) − 14.303 4. TP 4/S(P4) − 18.409

are as follows:

• C₁ (production costs)—1;
• C₂ (dimension accuracy Ø16^+0.018 mm)—2;
• C₃ (dimension accuracy Ø24 ± 0.2 mm)—3;
• C₄ (length accuracy 10 ± 0.2 mm)—4;
• C₅ (production time)—5;
• C₆ (roughness Ra 0.8 μm)—6.

The lowest value of the calculated technological processes with the indicator S(P_m) shows that the rational technological process is the one with the lowest calculated value, namely TP 1/S(P₁) = 12.520.

3.3.5. Stage 5—Selection of a Rational Technological Process

In order to demonstrate the flexibility and adaptability of the proposed method, Table 7 presents various combinations of the importance of the selected criteria. They are automatically generated in MS Excel and confirmed by the technical specialists at CERATIZIT Bulgaria AG.

The purpose of generating random combinations is to show that when customer requirements change (change in the weight of certain criteria)—for example, when the focus is on cost instead of quality—the final choice of a rational technological process may be different. For instance, a technological process that was previously ranked lower may now be the most appropriate if criteria associated with production costs and time are given more weight. This demonstrates how flexible the approach is and how it can be used practically when making decisions in light of shifting production objectives and limitations. The table is for demonstration purposes and also represents an example of the sensitivity of the methodology and was not used as a tool in the experts’ assessment.

Presenting different options in tabular form contributes to a better understanding of the impact of each criterion on the final result and emphasizes the importance of correctly selecting and weighing the criteria involved in the evaluation of the technological process.

4. Discussion

FUCOM stands out with consistent and reliable results, even when repeated calculations are performed, making it suitable for analyzing complex production tasks. This makes it suitable for integration into automated and digitized decision-making systems. Its potential for generalization is significant, as it can be adapted to various sectors, from mechanical engineering and logistics to energy and healthcare. With the proper calibration of criteria and combination with other MCDM methods, FUCOM can become a universal tool for evaluating and optimizing technological processes. The limitations of the method include its dependence on expert assessments, difficulties using the mathematical model when there are a large number of criteria, and the need for normalization when working with different types of data. Therefore, although it has its limitations, the method has a solid mathematical basis and potential for wide application in solving complex multi-criteria problems [9].

The results of the experimental studies and the application of the FUCOM method for selecting a rational technological process show that the method provides a high degree of objectivity and consistency in determining the priorities of different criteria. By applying this in the analysis of different technological processes for the manufacture of a single part, the subjectivity of expert choices and assessments is minimized, and the final decision is based on a mathematically sound model that ensures complete consistency between the weights of the criteria.

The analysis of the four technological processes considered (TP N° 1—CNC_53; TP N° 2—C11; TP N° 3—C11_47; and TP N° 4—STAMA) showed that TP N° 1 occupies a leading position in the ranking, given the weights (Table 5) of the specific criteria determining the part that was considered, as this offers an optimal balance between accuracy, roughness, production costs, and manufacturing time. However, in order to show the method’s adaptability to various evaluation priorities, Table 7 displays randomly generated combinations of criterion weights. This shows that varying the weights that different combinations of criteria can result in different final rankings of technological processes, reflecting the actual dynamics of production decisions based on the particular situation’s goals and constraints.

These results confirm that the FUCOM method can be successfully used in a production environment to evaluate technological processes, providing an opportunity for the clear and justified ranking of alternatives even in the presence of multiple and conflicting criteria.

Compared to other multi-criteria methods, such as AHP and BWM, FUCOM was chosen by the authors because of its advantages in terms of the number of expert comparisons required and the achievement of complete consistency. This makes the method more effective and easier to apply in real production conditions, where time and resources are often limited.

Despite the positive results, some limitations of the study should be noted. Determining the weights of the criteria through expert assessments based on production needs and customer requirements remains somewhat subjective and can be hindered by lack of experience, poor training of experts, and unfamiliarity with the production process when starting a new job (when the human factor is involved, no matter how much expertise there is, there is always an element of subjectivity). This requires careful selection of expert technologists and the application of additional methods to verify the consistency and stability of the assessments. In order to increase the reliability of the results, future developments may combine FUCOM with quantitative approaches or automated evaluation algorithms to limit the influence of the subjective factor.

Small changes even between two criteria lead to a change in weights, as demonstrated by the examples presented in Table 7. As can be seen, could result in the selection of a different technological process, which shows an adequate response to the methodology.

In order to evaluate the impact of the final decision when changing weights and normalizing criteria, it is useful to combine the FUCOM method with other methods, such as TOPSIS or VIKOR, in future studies. Such hybrid models may result in systems for selecting technological solutions that are even more precise and flexible.

5. Conclusions

This study demonstrates the applicability of the MCDM method FUCOM for evaluating and selecting a rational technological process in real production conditions. Through the experimental measurements and following analysis, the results were validated in an industrial environment, confirming that the method can be used effectively in decision-making related to the selection of technological processes in mechanical engineering production. The results obtained show good consistency between the mathematically calculated estimates and the actual quality and efficiency indicators observed in production.

The FUCOM method reveals significant potential for evaluating technological processes by allowing for multiple criteria to be considered simultaneously. Its high consistency in determining the weights of the criteria and the reduced number of expert comparisons make it suitable for practical application in companies seeking a balance between accuracy, quality, cost, and productivity.

Adding LWL to roughness control is an effective parameter for preventing overprocessing. The approach marks areas where unnecessarily smooth surfaces and overly precise dimensions are generated without improving the functionality of the product. This ensures balanced quality and a measurable reduction in cost.

The main advantages of the method include the following:

• Ensuring full consistency between the weights of the criteria;
• Minimizing the cognitive overload on experts;
• Transparency and easy traceability of results;
• Possibility of integration with other methods of multi-criteria analysis.

Nevertheless, there are some restrictions regarding the use of FUCOM. The approach depends on expert evaluations, which are subjective to some degree despite their limited quantity. Furthermore, if the scales are not properly defined, the normalization of quantitative criteria may result in distortions and necessitates careful preparation.

The study suggests a number of potential opportunities for further research, including the following:

• Extending the application of the FUCOM method to a wider range of technological processes in mechanical engineering, such as additive manufacturing;
• Combining FUCOM with other MCDM methods (e.g., TOPSIS, VIKOR, AHP) to build hybrid models for more sustainable and sensitive decision-making;

The FUCOM method demonstrates effectiveness, flexibility, and practical applicability in the assessment of technological processes in mechanical engineering. It provides a stable analytical basis for objective and transparent decision-making, and its application in the assessment of technological processes would contribute to the optimization of production processes.