Real Case Application Research Articles

Industry 4.0 technology implementation in manufacturing: a selection method and real case applications

The pressing necessity for digitalisation in industrial plants, driven by Industry 4.0 national initiatives and heightened global competition, underscores the urgency for companies to initiate digital transformation projects. Despite this urgency, the academic literature lacks comprehensive guidance on models specifically dedicated to the selection of digital technologies. This article addresses this gap by proposing a multi-criteria decision-making model, grounded in a methodological framework, for the systematic selection of digital technologies in the manufacturing sector. The proposed model combines fuzzy logic and the analytic hierarchy process (AHP) and incorporates a well-established classification of digital technologies. The model is able to select the single best candidate technology as well as the best candidate group of technologies that share the same purpose. In this way, the model tries to capture the interconnection element that is at the core of the digitalisation concept. To test its validity, the model was applied in two manufacturing companies operating in distinct production sectors. One of these companies was undergoing a digitalisation process in its plants, providing an additional basis for comparing the results of the proposed model.

Sustainable site planning for new launch facilities at spaceports

The New Space industry not only demands expanded spaceport capacity but also careful consideration of the environmental impact of ground infrastructure development. Focusing on the Alcântara Space Center in Brazil (CEA - Centro Espacial de Alcântara), a data-driven site selection framework is presented, emphasizing cost-effectiveness and environmental compliance. When applied to the CEA case, this framework incorporates expert preferences and aligns Geographic Information System-Multi-Criteria Decision Analysis (GIS-MCDA) with the development of launch site infrastructure and the assessment of environmental impact. The real case application results in robust ranking outcomes, despite some level of divergence amongst the experts. Contributions include expert clustering for sensitivity analysis, insights into spaceport impact on surrounding communities, and the proposal of an easily deployable decision-making process. The framework exhibits flexibility and can be adapted to different variations of the spaceport locational problems. Recommendations include the implementation of advanced geotechnical assessment protocols, enhanced quantification methodologies for environmental variables, and the utilization of comprehensive risk assessment techniques. These measures hold the potential to provide valuable insights for research and regulatory efforts within this rapidly evolving domain.

Review of Biomass Gasification and Pyrolysis Process Based on Long Short-Term Memory Network

Abstract. With the global energy crisis become escalated day by day, finding an effective way of exploring the renewable energy has become an important direction as its widely resources, comparatively low environment impact. This paper has concluded the biomass pyrolysis and gasification process based on long- and short-term memory (LSTM) method, including the LSTM principle, current application, faced challenges and future development. LSTM as a kind of special recurrent neural network, which could effectively solve the gradient vanish and explosion problem in time-series data. Research shows that, LSTM has an excellent performance in terms of reaction condition and optimization, which could be able to capture the relationship between complex variables, the pyrolysis efficiency could be improved, and prediction accuracy could over 90%. During the data preprocessing and model training stage, it is vital to make sure the quality of data, including data normalisation, outlier handling and time series splitting, etc. Through these steps, a better prediction accuracy could be achieved by LSTM. At the same time, the real application cases of LSTM used in biomass pyrolysis process have been discussed, it is shown that the combination of LSTM and other machine learning techniques could obviously improve the system stability and product quality. In addition, this paper has also looked forward to the enhancement of model explainability and the prospect of uncertainty quantification, which is helpful to improve the LSTM application and practical value.

Analysis and comparison of high-performance computing solvers for minimisation problems in signal processing

Several physics and engineering applications involve the solution of a minimisation problem to compute an approximation of the input signal. Modern hardware and software use high-performance computing to solve problems and considerably reduce execution time. In this paper, different optimisation methods are compared and analysed for the solution of two classes of non-linear minimisation problems for signal approximation and denoising with different constraints and involving computationally expensive operations, i.e., (i) the global optimisers divide rectangle-local and the improved stochastic ranking evolution strategy, and (ii) the local optimisers principal axis, the Limited-memory Broyden, Fletcher, Goldfarb, Shanno, and the constrained optimisation by linear approximations. The proposed approximation and denoising minimisation problems are attractive due to their numerical and analytical properties, and their analysis is general enough to be extended to most signal-processing problems. As the main contribution and novelty, our analysis combines an efficient implementation of signal approximation and denoising on arbitrary domains, a comparison of the main optimisation methods and their high-performance computing implementations, and a scalability analysis of the main algebraic operations involved in the solution of the problem, such as the solution of linear systems and singular value decomposition. Our analysis is also general regarding the signal processing problem, variables, constraints (e.g., bounded, non-linear), domains (e.g., structured and unstructured grids, dimensionality), high-performance computing hardware (e.g., cloud computing, homogeneous vs. heterogeneous). Experimental tests are performed on the CINECA Marconi100 cluster at the 26th position in the “top500” list and consider several parameters, such as functional computation, convergence, execution time, and scalability. Our experimental tests are discussed on real-case applications, such as the reconstruction of the solution of the fluid flow field equation on an unstructured grid and the denoising of a satellite image affected by speckle noise. The experimental results show that principal axis is the best optimiser in terms of minima computation: the efficiency of the approximation is 38% with 256 processes, while the denoising has 46% with 32 processes.

Synthesis and Characterization of Rebondable Polyurethane Adhesives Relying on Thermo-Activated Transcarbamoylation.

Dynamic polymer networks combine the noteworthy (thermo)mechanical features of thermosets with the processability of thermoplastics. They rely on externally triggered bond exchange reactions, which induce topological rearrangements and, at a sufficiently high rate, a macroscopic reflow of the polymer network. Due to this controlled change in viscosity, dynamic polymers are repairable, malleable, and reprocessable. Herein, several dynamic polyurethane networks were synthetized as model compounds, which were able to undergo thermo-activated transcarbamoylation for the use in rebondable adhesives. Ethylenediamine-N,N,N',N'-tetra-2-propanol (EDTP) was applied as a transcarbamoylation catalyst, which participates in the curing reaction across its four -OH groups and thus, is covalently attached within the polyurethane network. Both bond exchange rate and (thermo)mechanical properties of the dynamic networks were readily adjusted by the crosslink density and availability of -OH groups. In a last step, the most promising model compound was optimized to prepare an adhesive formulation more suitable for a real case application. Single-lap shear tests were carried out to evaluate the bond strength of this final formulation in adhesively bonded carbon fiber reinforced polymers (CFRP). Exploiting the dynamic nature of the adhesive layer, the debonded CFRP test specimens were rebonded at elevated temperature. The results clearly show that thermally triggered rebonding was feasible by recovering up to 79% of the original bond strength.

A knowledge-data dually driven paradigm for accurate identification of key blocks in complex rock slopes

Accurate identification and effective support of key blocks are crucial for ensuring the stability and safety of rock slopes. The number of structural planes and rock blocks were reduced in previous studies. This impairs the ability to characterize complex rock slopes accurately and inhibits the identification of key blocks. In this paper, a knowledge-data dually driven paradigm for accurate identification of key blocks in complex rock slopes is proposed. Our basic idea is to integrate key block theory into data-driven models based on finely characterizing structural features to identify key blocks in complex rock slopes accurately. The proposed novel paradigm consists of (1) representing rock slopes as graph-structured data based on complex systems theory, (2) identifying key nodes in the graph-structured data using graph deep learning, and (3) mapping the key nodes of graph-structured data to corresponding key blocks in the rock slope. Verification experiments and real-case applications are conducted by the proposed method. The verification results demonstrate excellent model performance, strong generalization capability, and effective classification results. Moreover, the real case application is conducted on the northern slope of the Yanqianshan Iron Mine. The results show that the proposed method can accurately identify key blocks in complex rock slopes, which can provide a decision-making basis and rational recommendations for effective support and instability prevention of rock slopes, thereby ensuring the stability of rock engineering and the safety of life and property.

Analysis and Modelling for Industrial Energy Efficiency in the Cosmetics Industry: A Real Industrial Case in an Italian Manufacturing Company

Economic and environmental issues translating into energy costs and pollution within the production environment are increasingly attracting attention. Industrial Energy Efficiency (IEE) is gaining ever-higher importance within production environments. Since cosmetic consumers and companies are becoming increasingly sensitive to sustainability, the cosmetic field is working to reduce the environmental and social impact along the whole supply chain. Furthermore, IEE actions in business processes can lead to several medium- and long-term economic and environmental benefits. This paper is the first work in the literature presenting a real-case application of energy analysis and modeling to achieve better energy performances in a cosmetics production process. Thus, in the body of knowledge, it contributes by providing a real case of good practice to be benchmarked for future IEE interventions in cosmetics manufacturing processes. The work has been conducted by analyzing the production process’s energy consumption and developing an energy model of a selected machine (i.e., a turbo-emulsifier). The analysis and modeling performed aimed at assessing the different operational phases of the machine and evaluating the different behaviors of the data. Finally, the results allowed us to propose possible improvements to be applied to the production process to achieve better energy performances.

A LMOF/MIP paper-based chip and analysis of tetracycline in foodstuff with sample-to-answer performance

The development of high-performance specific sensors is promising for the rapid detection of harmful residues in animal-derived foods. Recently, luminescent metal-organic framework/molecularly imprinted polymer (LMOF/MIP) materials have been developed as ideal candidates for the analysis of harmful residues. Here, we reported a simple fabrication protocol of paper-based chip through in-situ growth of LMOF on a negatively charged modified filter paper, a paper-based molecularly imprinting layer (FP@BA-Eu@MIP) was thereafter successfully prepared via the boronate affinity-based controllable oriented surface imprinting strategy. The paper-based chips obtained were used to construct a rapid test strip of tetracycline (TC). After addition of TC, significant fluorescence changes on the surface of the FP@BA-Eu@MIP paper-based chip could be observed from blue to red via inner filter effect and photo-induced electron transfer under the excitation of 360 nm. The adsorption kinetics was explored in detail. The presented strip exhibited satisfied selectiveness and sensitivity with a limit of detection of 8.47 μg L−1 for TC. It was confirmed that LMOF/MIP as a biomimetic recognition module can play a crucial role in enrichment and fluorescence response. This study provided a real application case for an in-situ fabricated fluorescence paper-based chip in rapidly detecting harmful residues.

Optimizing green splits in high‐dimensional traffic signal control with trust region Bayesian optimization

AbstractCentralized traffic signal control has long been a challenging, high‐dimensional optimization problem. This study establishes a simulation‐based optimization framework and develops a novel optimization algorithm based on trust region Bayesian optimization (TuRBO), which can efficiently obtain an approximate optimal solution to the high‐dimensional traffic signal control problem. Local Gaussian process (GP), trust region, and Thompson sampling are employed in the TuRBO and contribute considerably to performance in terms of computational speed, solution quality, and scalability. Empirical studies are carried out using data from Mudanjiang and Chengdu, China. The performance of TuRBO is compared with that of Bayesian optimization (BO), genetic algorithm and random sampling. The results show that TuRBO converges the fastest because of its ability to balance exploration and exploitation through the trust region and Thompson sampling. Meanwhile, because TuRBO enables more efficient exploitation through the local GP, the solution quality of TuRBO outperforms others significantly. The average waiting time achieved by TuRBO was 2.84% lower than that achieved by BO. Finally, the method has been successfully extended to a large network with 233‐dimensional spaces and 122 signalized intersections, demonstrating that the developed methodology can deal with high‐dimensional traffic signal control effectively for real case applications.

A novel prognostics solution for accurate identification of degradation patterns in turbo machines with variable observation window

The degradation of a system is a time bound phenomenon, which leads to the deterioration of turbomachinery, in terms of performance and reliability. If undetected and not acted upon in time, this could also lead to sudden system failure, resulting in unplanned unit downtime and maintenance. Unplanned downtime of a turbomachine leads to severe production loss for the end customer and consequent economic damages. Early detection of a degradation pattern would provide the customer with the opportunity to timely carry out corrective actions, preventing an unscheduled down time. The paper evaluates degradation identification methodology currently known from literature and finds them not accurate enough for general purpose application required by the solution. The paper discusses a novel methodology which can accurately detect degradation patterns of timeseries data. Critical features of this methodology are novel time-based correlation enabled regression model with variable observation window, autonomous training, and automatic adjusting capability to incorporate operating behavior change or physical system replacement. This leads to high accuracy, high generalization, and domain agnostic application capability. Moreover, particular focus is given to achieving high probability of detection and a low probability of false alarm. The paper demonstrates the performance achieved by the methodology when applied to the field of prognostics and diagnostics of IoT connected turbomachines through 50+ real application cases.

The Art of Finding the Right Drug Target: Emerging Methods and Strategies.

Drug targets are specific molecules in biological tissues and body fluids that interact with drugs. Drug target discovery is a key component of drug discovery and is essential for the development of new drugs in areas such as cancer therapy and precision medicine. Traditional in vitro or in vivo target discovery methods are time-consuming and labor-intensive, limiting the pace of drug discovery. With the development of modern discovery methods, the discovery and application of various emerging technologies have greatly improved the efficiency of drug discovery, shortened the cycle time, and reduced the cost. This review provides a comprehensive overview of various emerging drug target discovery strategies, including computer-assisted approaches, drug affinity response target stability, multiomics analysis, gene editing, and nonsense-mediated mRNA degradation, and discusses the effectiveness and limitations of the various approaches, as well as their application in real cases. Through the review of the aforementioned contents, a general overview of the development of novel drug targets and disease treatment strategies will be provided, and a theoretical basis will be provided for those who are engaged in pharmaceutical science research. SIGNIFICANCE STATEMENT: Target-based drug discovery has been the main approach to drug discovery in the pharmaceutical industry for the past three decades. Traditional drug target discovery methods based on in vivo or in vitro validation are time-consuming and costly, greatly limiting the development of new drugs. Therefore, the development and selection of new methods in the drug target discovery process is crucial.

Stochastic Modeling of Thin Mud Drapes Inside Point Bar Reservoirs With ALLUVSIM‐GANSim

AbstractModeling inclined fine‐scale mud drapes inside point bars, deposited on accretion surfaces during stages of low energy or slack water, is critical to modeling fluid flow in complex sedimentary environments (e.g., fluvial and turbidity flows). These features have been modeled using deterministic or geostatistical modeling tools (e.g., object‐, event‐, and pixel‐based). However, this is a non‐trivial task due to the need to preserve geological realism (e.g., connectivity within sedimentary features and facies hierarchy), while being able to condition the generated models to point data (e.g., well data). Generative Adversarial Networks (GAN) have been successfully applied to reproduce several large‐scale scenarios (e.g., braided rivers and carbonate reservoirs), yet their potential for capturing small‐scale and hierarchical features remains largely unexplored. Here, we propose a geo‐modeling workflow for fast modeling of small‐scale conditional mud drapes based on ALLUVSIM and GANSim. Initially, improved ALLUVSIM produces realistic unconditional models of mud drapes along accretionary surfaces, serving as GAN training data. GANSim is then employed to achieve conditioning to well data and probability maps derived from geophysical modeling. Finally, temporal pressure data observed in wells are further conditioned via a Markov chain Monte Carlo sampling method. The proposed geo‐modeling workflow is validated in a two‐dimensional synthetic example as the pre‐trained generator extracts mud‐drapes‐features and generates multiple facies realizations conditioned to diverse information. A field application example in a modern meandering river verifies the effectiveness and practicability of the proposed workflow in real case application examples. The application examples illustrate the potential of the proposed method to predict mud drapes inside point bar reservoirs.

ENTREPRENEURSHIP CASE OF PROFESSORS

Objective: The paper aims to analyse the stages and problems involved in the creation of a technology-based university spin-off by providing a real case study. Theoretical Framework: The main concepts supporting the research are discussed, as well as the theoretical framework of reference. It addresses the stages of the process of setting up a university technology-based company, as well as the specific problems that it involves. All issues are presented in a real context. Method: The methodology adopted for this research is based on a case study for teaching application. To this end, information is gathered through observation, direct interviews with the company under study, public documentation, and theoretical review through scientific databases of impact. Results and Discussion: The results were reflected in the development of a university teaching activity with guidelines for the teachers coordinating its implementation, in terms of objectives and reference bibliographic material to be used. It was contextualised in the light of the theoretical framework, highlighting the implications, basically at the level of reinforcement and acquisition of competences. Research Implications: The paper has essentially practical implications, providing information that reinforces the existing scientific literature on how the decision-making process that leads to the setting up of a technology-based university spin-off takes place and the problems associated with it. This allows a vision that enables a more effective and efficient execution of the entrepreneurial decision. Originality/Value: This is a real case of application in the university classroom, which allows students to assimilate the issues related to the process of entrepreneurship and, specifically, to the setting up of a technology-based university spin-off, dealing with its specific problems. Students' skills are strengthened with a view to their incorporation into the world of professional life.

Transactive energy based on virtual power plant and ancillary services: A real case application in Brazil after the Law 14.300/2022

This paper introduces a transactive energy model that incorporates a Virtual Power Plant composed of photovoltaic systems, batteries and hybrid systems within an ancillary services framework. In response to the enactment of Law 14.300/2022 in Brazil, which has led to diminished commercial incentives for the installation of Distributed Energy Resources, investors are compelled to explore alternative avenues to offset these losses. The paper explores the coordination of Distributed Energy Resources as a Virtual Power Plant for commercial purposes, simultaneously addressing technical challenges such as promoting voltage control, mitigating electric losses, and reducing power demand. Simulations conducted in this paper utilizing a real case system in Brazil demonstrate that the proposed approach holds promise. This is attributed to the flexibility afforded by charging the batteries, as well as the inverter's capacity in the Distributed Energy Resources to inject reactive power, thereby enhancing the overall value of the proposed business model. The simulations conducted reveal that economic benefits are notably attained by charging batteries during off-peak periods and discharging them during peak period. Furthermore, due to the significant integration of photovoltaic systems, charging during off-peak hours contributes to the mitigation of overvoltage issues and reduction of energy losses. It is important to highlight that, during peak periods, the optimization of commercial gains is contingent upon adhering to a prescribed 3-h limit for battery discharging. Notably, the prioritization of commercial gains during peak times has the potential to enhance power demand reduction, as evidenced in the case study presented in this paper.

An interpretable operating condition partitioning approach based on global spatial structure compensation-local temporal information aggregation self-organizing map for complex industrial processes

Effective and interpretable partitioning of operating conditions in complex industrial processes is crucial for ensuring the efficiency and reliability of emerging integrated production processes. However, contemporary operation–condition partitioning models fall short in this domain for two main reasons. First, the parameters related to purely data-driven methods fail to involve physically meaningful semantics, resulting in suboptimal accuracy and interpretability. Second, there exists a spatial–temporal coupling interaction among process variables that reflects physically meaningful spatial–temporal correlations, further impeding effective interpretation. To address these challenges, an interpretable operation–condition partitioning method based on a Global Spatial Structure Compensation-Local Temporal Information Aggregation Self-Organizing Map is proposed. This approach formulates the operation–condition partitioning problem as a spatio-temporal feature extraction based on the operation–condition partitioning problem. First, a local separable aggregation (LSA) model based on the informer backbone is introduced to partition industrial variables into interpretable time-local groups. This ensures that the model can learn the underlying factors of the condition’s occurrence, thereby providing a reliable basis for interpretable root cause analysis. Next, a Global Spatial Feature Compensator (GSFC) is introduced that leverages the process mechanism knowledge and graph network to establish explicit spatial interrelationships among time-local groups in the LSA algorithm, thereby enhancing the model's inferential capabilities and security assurance. Subsequently, a spatiotemporal self-organizing map is designed to integrate the aforementioned spatial and temporal features, facilitating precise operation–condition partitioning decisions. Finally, the effectiveness and understandability of the proposed method are validated through comprehensive experiments in a real industrial application case study.

Dynamic inference for left behind probabilities on congested metro routes

ABSTRACT Passengers left behind is an important measure to describe the degree of congestion in metro systems. Note that passengers’ left behind probabilities are different for their different tap-in times. This paper proposes a methodology for inferring these dynamic probabilities on congested metro routes using automated data. The EM algorithm is used to compute the maximum likelihood estimators of passengers’ dynamic boarding probabilities, and then formulas for estimating dynamic left behind probabilities are presented based on the estimated boarding probabilities. Monte Carlo simulations and a real case application show the effectiveness of the proposed method.

Belt rotation in pipe conveyors: Development of an overlap monitoring system using digital twins, industrial Internet of things, and autoregressive language models

Belt rotation is a critical failure mode in pipe conveyors. However, overlap position monitoring remains unexplored. Existing monitoring solutions are expensive and lack scalability, primarily relying on detecting angle variations beyond predefined thresholds. We developed a digital twin of a low-cost test rig for measuring belt rotation using 3D prototyping, infrared distance sensors, and a 3-axis accelerometer. We implemented a controller-responder industrial Internet of Things network using ESP-NOW protocol. Applying a minimum viable prompt, we fine-tuned an autoregressive language model for predicting the overlap position. We then developed a software application for cloud computing and local storage of measurements. Additionally, we developed a human–machine interface for future real-case applications. The bias and precision for the predicted angle (0.2294° ± 11.8822°), angular speed (0.1630 ± 0.7344 RPM), number of laps (0.0001336 ± 0.0448 laps), diameter (–0.1984 ± 6.3743 mm), and outer-edge side (100 % agreement) were suitable for the intended use.

Artificial Intelligence and Forensic Genetics: Current Applications and Future Perspectives

The term artificial intelligence (AI) was coined in the 1950s and it has successfully made its way into different fields of medicine. Forensic sciences and AI are increasingly intersecting fields that hold tremendous potential for solving complex criminal investigations. Considering the great evolution in the technologies applied to forensic genetics, this literature review aims to explore the existing body of research that investigates the application of AI in the field of forensic genetics. Scopus and Web of Science were searched: after an accurate evaluation, 12 articles were included in the present systematic review. The application of AI in the field of forensic genetics has predominantly focused on two aspects. Firstly, several studies have investigated the use of AI in haplogroup analysis to enhance and expedite the classification process of DNA samples. Secondly, other research groups have utilized AI to analyze short tandem repeat (STR) profiles, thereby minimizing the risk of misinterpretation. While AI has proven to be highly useful in forensic genetics, further improvements are needed before using these applications in real cases. The main challenge lies in the communication gap between forensic experts: as AI continues to advance, the collaboration between forensic sciences and AI presents immense potential for transforming investigative practices, enabling quicker and more precise case resolutions.

Noninvasive system for weight estimation in cactus crops: A YOLOv5-decision tree approach based on interval type-2 fuzzy sets

This study proposes a noninvasive system to estimate the weight of a prickly pear cactus (Opuntia ficus-indica), which combines a model based on deep learning to detect and estimate its area and a model based on supervised learning and developed under a type-2 interval fuzzy sets approach to estimate its weight. YOLOv5 was used for detection, which achieved results with an accuracy of 0.999, recall of 0.999, mAP_0.5 of 0.995, and mAP_0.5:0.95 of 0.990. Area estimation is performed based on the object's coordinates in the image. To perform the weight estimation task, a comparative analysis of models based on supervised learning was performed to identify the model that best described the response variable, selecting a decision tree model. The training and adjustment process included the uncertainty in a weight evaluation system using a granatary scale to enhance the weight estimation capability. Using a trapezoidal membership function, the type-2 interval fuzzy set approach was used to convert the collected data into fuzzy numbers. The model findings for the weight estimation task were also favorable, with a model-explained variance R2 of 0.98, a root mean square error (RMSE) of 10.02, and a mean absolute error (MEA) of 6.33 g. Finally, both models were incorporated into a graphical user interface to facilitate their use in the real weight estimation process. A real case application with the proposed system demonstrated its effectiveness in detecting and estimating weight dynamically with an error of 9 g. Therefore, its viability as an alternative system for weight estimation in cactus crops was demonstrated. Furthermore, a comparative analysis demonstrated the superiority of the proposed hybrid approach over similar approaches in the literature.

Design and analysis of Cpm and Cpmk indices for uncertainty environment by using two dimensional fuzzy sets

Process capability analysis (PCA) is an important stage to check variability of process by using process capability indices (PCIs) that are very effective statistics to summarize process’ performance. Traditional PCIs can produce some incorrect results and declare misinterpretation about process’ quality if the process includes uncertainties. Additionally, definitions of process’ parameters with exact values is not possible when there are uncertainty caused by measurement errors, sensitivities of measuring instruments or quality engineers’ hesitancies. Although the fuzzy set theory (FST) has been successfully used in PCA, it is the first time to use of Pythagorean fuzzy sets (PFSs) to model uncertainties of process more than traditional fuzzy sets in PCA. Since the PFSs has two-dimensional configurations by defining membership and non-membership values, they also have a huge ability to model uncertainty that arises from the human’s thinking and hesitancies, and has brought flexibility, sensitivity and reality for PCA. In this paper, specification limits (SLs), mean (μp), standard deviation (σ) and target value (T) main parameters of PCIs have been analyzed by using PFSs and Pythagorean fuzzy process capability indices (PFPCIs) for two well-known PCIs such as ( C ˜ pm ) and ( C ˜ pmk ) have been derived. The Pythagorean ( C ˜ pm ) and ( C ˜ pmk ) indices have also been applied and tested on some numerical examples based on real case applications from manufacturing industry. The obtained results show that PFPCIs provide wider knowledge about capability of process and to obtain more realistic results. As a result of considering all possibilities about the process, it has been concluded that the process is incapable. In light of this information, the results obtained using different fuzzy set extensions for (Cpm) and (Cpmk) indices can be compared.