The research, titled “Enhancing industrial robot selection through a hybrid novel approach: integrating CRITIC-VIKOR method with probabilistic uncertain linguistic q-rung orthopair fuzzy,” was conducted by authors such as Sumera Naz, Muhammad Muneeb ul Hassan, and Atif Mehmood, in collaboration between institutions like the University of Education (Lahore, Pakistan) and Zhejiang Normal University (China). The study focuses on choosing industrial robots when numerous variables are involved, and not all are certain. To tackle this, it proposes a model that uses two analysis techniques (CRITIC and VIKOR) combined with a system capable of handling imprecise information, expressed through linguistic terms and probabilities. In other words, this model helps evaluate different robots, considering vague or uncertain data, thus providing a more flexible and clearer method for selecting the machine best suited to a company’s needs.
Managing Uncertainty in Industrial Robot Selection with CRITIC-VIKOR
Selecting an industrial robot using the CRITIC-VIKOR method is a strategic step for many modern companies, as robots play a key role in automating production processes and optimizing resources. However, the multiplicity of attributes to consider makes the choice complex. Experts must deal with heterogeneous factors ranging from load capacity to the quality of the human-machine interface, as well as more nuanced criteria such as ease of programming or the level of service offered by the supplier. In addition to purely numerical parameters—such as maximum achievable speed or positioning accuracy—it is often necessary to evaluate qualitative aspects. Examples include how easily a robot can be integrated into existing production environments, its flexibility in adapting to future tasks not yet fully defined, or the perceived reliability of the manufacturer. Such aspects are not simple to quantify because the human mind translates impressions into linguistic terms that, if transposed into a decision analysis, risk losing part of their semantic richness.
This scenario highlights the need for an approach that values the uncertainty inherent in human judgments. In traditional methods for choosing among alternatives, data are often converted into numbers that do not fully reflect the nuanced nature of the decision-makers’ evaluations. When an expert describes a robot’s accuracy as “good” instead of providing an exact number, they are not expressing a fixed value, but a multidimensional, variable perception influenced by experience and context. Linguistic uncertainty should not be viewed as a limitation but rather as a source of information that can provide a more realistic perspective of the decision-maker’s thoughts.
For this reason, representing information through fuzzy linguistic sets enriched by probabilities (as in the case of PULq-ROFS) allows balancing numerical precision with cognitive flexibility. The probabilities associated with linguistic terms make it possible to capture the frequency or credibility a decision-maker assigns to a particular expression. For example, if a judgment on a robot’s repeatability is “mainly good, sometimes very good,” two linguistic intervals with related probabilities can be used, indicating more precisely how often the decision-maker expects the robot to operate at a given qualitative level. Moreover, the q parameter allows adjusting the degree of uncertainty, adapting the model to the problem’s specific needs. Situations with greater complexity or very divergent expert opinions require finer granularity, while simpler scenarios can be managed with less complex settings.
Such a representation is particularly advantageous in highly variable production environments, where required robot characteristics can rapidly change as lines evolve, demand shifts, or corporate strategies adapt. Using PULq-ROFS does not merely provide a new formalism but opens the possibility of better integrating human perceptions, probabilistic estimates, and objective measurements. This delineates a decision-making process that is more consistent with industrial realities. Robot selection thus moves beyond a mere table of technical specifications, becoming an evaluative journey where uncertainty complexity, incomplete data, and diverging expert judgments find a rigorous yet flexible channel of expression.
CRITIC-VIKOR: A Hybrid Approach for Industrial Robot Selection
In the proposed methodology, combining the CRITIC and VIKOR methods within the PULq-ROFS context addresses the need to go beyond the limits of a purely subjective multicriteria analysis. Traditional approaches, based solely on human evaluation, risk being unbalanced: a decision-maker may assign excessive importance to criteria that are not truly crucial overall, or neglect objective parameters because they are less eye-catching. This creates a potential gap between individual perception and the actual discriminating value of each attribute.
The CRITIC method directly addresses this issue, bringing order to the complexity of criteria.
CRITIC (Criteria Importance Through Intercriteria Correlation) does more than simply compile a list of criteria with preassigned weights. Instead, it analyzes the data to measure their variability and correlation. If an attribute shows high variability among alternatives and provides information not overlapping with other criteria, it means this attribute has strong discriminating power. In practice, CRITIC assigns greater weight to those indicators that truly differentiate the alternatives. Conversely, if a criterion is strongly correlated with others and does not contribute new information, its weight will be lower. This procedure, based on statistical and mathematical measures, reduces the influence of arbitrary judgments by decision-makers and produces a more objective set of weights.
Take a concrete example: imagine evaluating four robots (M1, M2, M3, M4) based on attributes like load capacity, programming flexibility, repeatability, cost, and other technical aspects. Without the CRITIC method, a decision-maker might, due to personal preference or unmeasured perception, assign a much higher weight to accuracy compared to programming flexibility. However, CRITIC might discover that accuracy shows very similar trends among all robots (i.e., their values are all quite close), while programming flexibility varies significantly and is not closely correlated with other criteria. Consequently, CRITIC will assign a higher weight to flexibility because it can “make the difference” in the final choice.
Once the weights are defined through CRITIC, VIKOR comes into play. This method ranks and selects solutions capable of handling situations where criteria conflict with each other. VIKOR was developed to provide a compromise solution that lies as close as possible to the ideal. Its crucial aspect is the ability to simultaneously consider maximizing collective utility (the overall benefit of the attribute group) and minimizing individual regret (how far an alternative deviate from the best attribute overall). In other words, VIKOR does not simply choose the globally best candidate; it offers a balanced vision that acknowledges potential conflicts between criteria. For instance, one robot might excel in precision but be less cost-effective and less versatile. VIKOR identifies a point of balance, a solution that is not only optimal in a strict sense but also better satisfies all-around interests.
By integrating these methods, CRITIC-VIKOR addresses the need to overcome the limits of purely subjective multicriteria analysis in industrial robot selection. On one hand, CRITIC eliminates or reduces initial bias by defining more coherent weights based on the data’s informational structure. On the other hand, VIKOR, using these weights and probabilistic fuzzy evaluations, calculates the compromise solution that best respects the identified priorities. The result is a decision-making process with multiple layers of robustness: data are first interpreted in probabilistic fuzzy form (PULq-ROFS), then weighted objectively (CRITIC), and finally ranked following a compromise approach (VIKOR). This three-pronged method ensures that the robot selection results from a balanced, rational evaluation that carefully considers the various facets of uncertainty.
Practical Application of CRITIC-VIKOR in Robotic Selection
In actual business scenarios, identifying the ideal industrial robot for a specific production application cannot be limited to a simple comparison of technical parameters. Although robots often share basic characteristics such as degrees of freedom, speed, load capacity, and precision, each one possesses distinctive features—both technical and qualitative—that make direct comparisons difficult.
In this hypothetical scenario, consider four industrial robots (M1, M2, M3, and M4), evaluated based on four fundamental attributes: load capacity (N1), programming flexibility (N2), repeatability (N3), and cost (N4). These criteria, chosen in collaboration with technical consultants and plant managers, reflect both objectively measurable aspects—such as the ability to handle heavy loads or maintain a high degree of accuracy over time—and more subjective elements, like how easily an operator can update the robot’s settings or the perception of long-term economic convenience.
Initial data come from expert assessments, not expressed through certain numbers but with uncertain, probabilistic linguistic terms. For example, an expert might describe the programming flexibility of M1 as “mostly high, with a moderate probability of being considered very high,” whereas for M2 they might say “flexibility varies between medium and high, but rarely very high.” These expressions are transformed into PULq-ROFS sets, encoding the expressed linguistic intervals and the associated degree of uncertainty. This approach captures the cognitive complexity behind human judgments, integrating both the variability of individual decision-makers and the possibility that a judgment may not always be the same in all contexts.
After obtaining a PULq-ROFS evaluative framework, an aggregation operator like the PULq-ROF Weighted Average is applied. This step allows the combination of judgments from multiple experts into a single composite evaluation, considering the respective weights of the decision-makers or information sources. The result is a decision matrix representing a shared perception, integrating the viewpoints of various stakeholders (managers, line technicians, maintenance personnel, logistics experts).
The CRITIC method is then applied to the aggregated result to determine the attributes’ weights objectively. Suppose the CRITIC calculations show that programming flexibility (N2) is a highly discriminating criterion compared to the other characteristics, thus assigning it a higher weight. This step highlights that, among differences between M1, M2, M3, and M4, the way the robot can be reprogrammed or adapted to process changes significantly impacts the final choice.
At this point, the VIKOR method is used on the processed data to identify the alternative that best fits the compromise between maximizing benefits and minimizing regret, based on CRITIC’s weights and the probabilistic fuzzy evaluations of the robots. VIKOR returns a final ranking. For example, in the presented case, M2 might emerge as the solution closest to the ideal compromise, showing an excellent balance between objective criteria like repeatability and more qualitative aspects like programming flexibility.
Robustness Analysis of the CRITIC-VIKOR Method for Industrial Robots
The concrete application and initial results of the proposed method lead to an important question: how robust and resilient is the model when facing variations in parameters or input information? To explore this, it is essential to examine the model’s sensitivity to changes in some fundamental parameters, with special attention to the q parameter, which regulates the degree of granularity and uncertainty within q-rung orthopair fuzzy sets.
The q parameter acts as a “precision regulator” in q-rung orthopair fuzzy sets. When q takes on higher values, the system can more finely distinguish differences between alternatives, capturing and representing subtler nuances of linguistic uncertainty. On the other hand, lower q values result in less granularity, simplifying the analysis but potentially losing some of the informative complexity.
In-depth investigation has shown that varying q—for example, by increasing it from 1 to 35 at irregular intervals (e.g., using only odd values)—does not overturn the ranking results produced by VIKOR integrated with PULq-ROFS and CRITIC. Certainly, small differences may be noted in the margin between alternatives or in the intensity of intermediate scores, but the final order of preferences remains substantially unchanged. This finding is significant: it indicates that the methodology is not fragile. A small perturbation in uncertainty parameters does not reverse the conclusions. On the contrary, the model shows sufficient robustness to maintain stable key choices and the ranking of alternatives, regardless of how finely uncertainty is represented.
Another crucial aspect of sensitivity analysis is how the method reacts to the addition of new experts or the removal of an information source. In real environments, data and evaluations are not always available in a complete and stable form: sometimes an expert is no longer available, or it may be necessary to incorporate the opinion of a new specialized technician. Thanks to the use of PULq-ROF aggregation operators and the objective weight calculation (CRITIC), the proposed method can quickly update the analysis without starting from scratch. A recalculation of the PULq-ROFWA aggregation is sufficient, followed by updated weights (CRITIC) and a new ranking (VIKOR). The approach’s modular structure ensures easy adaptation to changes in the context.
Moreover, the combined use of CRITIC and VIKOR in the PULq-ROFS framework further enhances the decision-making process’s resilience. CRITIC ensures that attribute weights are not chosen arbitrarily, but instead emerge from statistical data analysis. VIKOR does not seek the simply “best” alternative, but one that is closest to a compromise solution, reducing the ranking’s sensitivity to extreme conditions. This methodological arrangement creates an internal balance: if one attribute is evaluated less stably, other criteria and the entire weighting process help limit the impact of that fluctuation on the result.
CRITIC-VIKOR vs. Other Methods: Advantages in Industrial Robot Selection
The above analysis highlights how integrating PULq-ROFS with CRITIC and VIKOR yields a particularly solid approach. To fully understand its value, it is useful to compare it with some well-known methods from the multicriteria decision literature.
In the field of robotic selection and, more broadly, evaluating complex alternatives, there are many established approaches like TOPSIS, MABAC, CODAS, or EDAS, including fuzzy or probabilistic variants. While these methods are extremely useful, they sometimes have limitations when it comes to simultaneously handling complex linguistic uncertainties, probability associated with terms, and a weighting structure not solely based on subjective judgment. For example, a method like TOPSIS generally works with distances from ideal solutions but does not inherently integrate the logic of linguistic probability or the adaptability of the q parameter. Similarly, MABAC or CODAS can provide effective classifications but may not always offer the same flexibility in expressing uncertainty, nor the ability to extract attribute weights truly objectively, as CRITIC does.
The fuzzy variants introduced in some of these methods, such as PUL-TOPSIS, PL-MABAC, or PUL-CODAS, represent a step forward in incorporating linguistic uncertainty. However, these versions often remain tied to a single level of fuzzification or do not fully exploit the combined potential of a system that unites probabilities, q-rung orthopair sets, and objective weighting methods. In these approaches, determining weights may still depend largely on decision-makers’ discretionary choices, or the representation of linguistic uncertainty might remain partial.
The proposed strategy differs substantially. Using PULq-ROFS makes it possible not only to capture the vagueness and uncertainty inherent in judgments but also to model the probability with which certain linguistic intervals occur. This means the information is never impoverished; rather, it is enriched. Instead of reducing a verbal evaluation to a single central value, a probabilistic range of possibilities is maintained. This is particularly useful in situations where decision-makers cannot always guarantee the same perception, or where the robot’s operating conditions can change over time.
The use of the CRITIC method to determine weights adds further value compared to systems relying solely on subjective judgments or simple averages. CRITIC uses the information contained in the data to establish the relative importance of criteria, highlighting those that truly discriminate between alternatives. This reduces the risk of distortions tied to overly strong personal preferences, providing an objective basis for calculating weights.
Finally, integrating VIKOR adds another piece to the puzzle. Unlike other methods that aim straight for the globally best alternative, VIKOR favors a compromise solution that brings the system closer to a balance between overall benefits and minimal regrets. This makes the choice more resistant to conflicts between criteria and minor input changes, as demonstrated by the sensitivity analysis.
In summary, the proposed methodology not only fills gaps found in other approaches but also creates a harmonious ensemble of tools: PULq-ROFS manages the complexity of linguistic uncertainty, CRITIC extracts objective weights, and VIKOR leads to a compromise solution. This framework, seen from an industrial strategy perspective, allows identifying a robot that is not only “theoretically better” but also more aligned with operational needs and real constraints. Entrepreneurs, managers, and decision-makers can rely on a more comprehensive analytical framework, capable of reducing the influence of biases or incomplete information, offering a competitive advantage in tackling complex choices with greater confidence and clarity.
Conclusions
The results suggest that integrating CRITIC and VIKOR in a PULq-ROFS environment provides a solution capable of addressing the typical complexity of advanced industrial choices. The approach does not merely handle uncertainty but also reflects the real dynamics of conflicts between criteria, offering a solid tool for decision-makers and managers. Compared to the state of the art, this solution appears more balanced and versatile, especially where other methodologies focused on a single level of fuzzification, or a single probabilistic approach may risk oversimplifying the problem. From an industrial strategy perspective, such analytical capabilities make it possible to identify options that satisfy both operational needs and economic constraints, positioning the choice of the robot to support innovation and competitiveness. The interaction of quantitative methods and probabilistic linguistic representations proves to be a significant tool for those who must make strategic decisions in uncertain environments, showing how a hybrid perspective can become a stable reference point in selecting complex industrial systems.
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