Robot sensors process based on generalized Fermatean normal di ﬀ erent aggregation operators framework

: Novel methods for multiple attribute decision-making problems are presented in this paper using Type-II Fermatean normal numbers. Type-II Fermatean fuzzy sets are developed by further generalizing Fermatean fuzzy sets and neutrosophic sets. The Type-II Fermatean fuzzy sets with basic aggregation operators are constructed. The concept of a Type-II Fermatean normal number is compatible with both commutative and associative rules. This article presents a new proposal for Type-II Fermatean normal weighted averaging, Type-II Fermatean normal weighted geometric averaging, Type-II generalized Fermatean normal weighted averaging, and Type-II generalized Fermatean normal weighted geometric averaging

Abstract: Novel methods for multiple attribute decision-making problems are presented in this paper using Type-II Fermatean normal numbers. Type-II Fermatean fuzzy sets are developed by further generalizing Fermatean fuzzy sets and neutrosophic sets. The Type-II Fermatean fuzzy sets with basic aggregation operators are constructed. The concept of a Type-II Fermatean normal number is compatible with both commutative and associative rules. This article presents a new proposal for Type-II Fermatean normal weighted averaging, Type-II Fermatean normal weighted geometric averaging, Type-II generalized Fermatean normal weighted averaging, and Type-II generalized Fermatean normal weighted geometric averaging. Furthermore, these operators can be used to develop an algorithm that solves MADM problems. Applications for the Euclidean distance and Hamming distances are discussed. Finally, the sets that arise as a result of their connection to algebraic operations are emphasized in our discourse. Examples of real-world applications of enhanced Hamming distances are presented. A sensor robot's most important components are computer science and machine tool technology. Four factors can be used to evaluate the quality of a robotics system: resolution, sensitivity, error and environment. The best alternative can be determined by comparing expert opinions with the criteria. As a result, the proposed models' outcomes are more precise and closer to integer number a classical set, FS and IVFS. The Pythagorean neutrosophic interval-valued set (PyNSIVS) was developed by Smarandache et al. [20]. The single-valued NSS is discussed using context analysis [21] and medical diagnosis [22]. Ejegwa [23] extended the distance measures for IFSs, such as HD, ED normalized HD and normalized ED. Recently, Palanikumar et al. [24] discussed the MADM using spherical vague normal operators and their applications for the selection of farmers. Recently, Garg et al. [25] discussed the new notion of confidence levels-based cubic Fermatean fuzzy AOs and their application to MCDM problems. Chakraborty et al. [26] discussed the real life applications based on Fermatean fuzzy Bonferroni mean AOs for selecting optimal health care waste treatment technology. Zeb et al. [27] interacted with the Fermatean fuzzy soft AOs and their application in symptomatic treatment of COVID-19. Khan et al. [28] discussed the concept of generalized neutrosophic cubic AOs using MADM method.
In order to better understand Type-II Fermatean normal number (Type-II FNN) information, this study will expand on the idea of Type-II Fermatean information. We can get Type-II FN data from AOs. This paper consists of seven sections. In Section 2, you will find a concise overview of Type-II Fermatean information. A Type-II FNN MADM is discussed in Section 3 for different AOs. For various AOs, Type-II FNN MADM is discussed in Section 4. Section 5 presents an example, analysis, and discussion of a method based on Type-II FNN. In Section 6, a conclusion is presented. In decision making, we may have one problem 0 (τ t Ξ (ε)) 2 + (τ i Ξ (ε)) 2 + (τ f Ξ (ε)) 2 > 2.
Consequently, this work makes the following contributions: (1) It has been established that Type-II FNN satisfy some basic algebraic operations, such as associative, distributive and idempotent laws. (2) We introduced HD and ED for Type-II FNNs. In order to calculate the ED between two Type-II FNNs. Furthermore, there is a discussion regarding converting Type-II FNNs to NFNs. (3) It is demonstrated numerically how MADMs and AOs are applied in Type-II FNN to real-world scenarios. A Type-II FNN algorithm needs to be developed. We also need to determine a normalized decision matrix using a Type-II FNN algorithm. (4) In relation to Type-II FNWA, Type-II FNWG, Type-II GFNWA, and Type-II GFNWG. We determined the ideal value of each concept. (5) By using the Type-II FNWA, Type-II FNWG, Type-II GFNWA, and Type-II GFNWG operators, different ranking results of alternatives can be derived flexibly, so that a decision-maker may select the ranking result according to his or her preferences. Due to its strong flexibility, the method proposed in this paper is appealing. The results of the DM are based on integer δ.

Basic concepts
We discuss some common definitions in this section.

Distance measure for Type-II FNN
We discuss the distance between two Type-II Fermatean normal numbers (Type-II FNNs) in this section.
Proof. The proofs are straightforward.

Aggregation operators for Type-II FNN
We introduced the new AOs for Type-II FNN in this section.

Type-II FN weighted averaging (Type-II FNWA) operator
Proof. The mathematical induction method yields the proof.
Using Definition 3.3, we get Using induction, n 3, It holds for any k.

Type-II FN weighted geometric (Type-II FNWG) operator
We define the Type-II FN weighted geometric operator (Type-II FNWG) in this section.
Proof. The mathematical induction method yields the proof.

Type-II generalized FNWA (Type-II GFNWA) operator
We define the Type-II generalized FNWA operator in this section.
Proof. The mathematical induction procedure leads to the proof. Let us first demonstrate that, If n = 2, then By using Definition 3.3, we get, This theorem holds for any k ∈ N.

Type-II Generalized FNWG (Type-II GFNWG) operator
We define the Type-II generalized FNWG operator in this section. Theorem 4.5. Let Ξ p = (υ p , ς p ); τ t p , τ i p , τ f p be a collection of Type-II FNNs. Then, Proof. Mathematical induction leads to the proof. Let us first demonstrate that If n = 2, then By using Definition 3.3, we get Remark 4.2. If δ = 1, the Type-II GFNWG is converted to the Type-II FNWG.

Algorithm
This section contains the algorithm for Type-II FN information. The best option is chosen according to this algorithm.
Step 1: Enter the decision values for Type-II FN.
Step 2: Construct a normalized decision matrix.
Step 3: For each attribute e j in j , Step 4: Calculate the optimal values using both positive and negative Type-II FNN. Step 5: Calculate the HDs using two ideal values with each choice.
Step 6: Ranking the choices based on relative closeness values.
Step 7: In order to find the best solution to the problem, the output yields are determined by sup 0 * j .

Selection of robot sensors
Sensors on robots are used to monitor a robot's condition and environmental conditions by passing these signals to a controller. In robots, sensors are based on the sensory organs of humans. It is imperative that robots have extensive knowledge of their environment in order to function properly. It is possible to measure a physical quantity with a sensor. These inputs can be converted into electrical signals to be recorded and monitored. In order to choose the right sensor, one must consider the robot's use and the environment in which it is operating. Sensor that measure the same input can differ in accuracy, precision, and sensitivity. It is important to keep this in mind when designing robots. There is a wide array of inputs that sensors can measure. Among these are the environment, force, speed, pressure, temperature, vibration, rotation, imaging, light, biometrics, acceleration, current, orientation, gravity and speech. Automation of robotics requires sensors. It is not possible for a robot to function without them. As a robot interacts with its environment through sensors, it overcomes obstacles and obtains information from the environment. A sensor can be selected according to its characteristics, and the characteristics of a sensor determine the type of sensor to be used for a given application.
(1) Light sensor ( 1): The light sensor is a transducer that detects light and creates a voltage difference that reflects the intensity of light falling on it. Generally, robots use photovoltaic cells and photo resistors as light sensors. There are a few other kinds of light sensors in use, such as photo transistors and photo tubes. In robotics, the following types of light sensors are used: A photo resistor is a type of resistor that detects light. The resistance of a photo-resistor changes with the intensity of light. As the resistance of a photo resistor increases, a current passing through it decreases. A photo resistor can also be known as a light dependent resistor (LDR). A photovoltaic cell is a device that converts solar radiation into electricity. It is used for building solar robots. Photovoltaic cells are considered energy sources by themselves; however, they can be converted to sensors when combined with capacitors and transistors. (2) Proximity sensor ( 2): A proximity sensor detects nearby objects without physical contact.
Proximity sensors are simple devices. Electromagnetic radiation is transmitted by a proximity sensor, and a return signal is received and analyzed for interruptions by the receiver. Therefore, the amount of light a receiver receives from its surroundings can be used to detect nearby objects. In robotics, proximity sensors are used in the following ways: A transceiver transmits infrared (IR) light via LEDs, but if it encounters an obstacle, the beam of IR light is reflected back, which is captured by a receiver. In ultrasonic sensors, high frequency sound waves are sent out by a transmitter, and the received echo pulse indicates that an object has been interrupted. In robotic systems, ultrasonic sensors are used to measure distance. (3) Acceleration sensor ( 3): The acceleration sensor measures acceleration as well as tilt. Using an accelerometer, you can measure acceleration. An accelerometer is affected by two types of forces: An object's frictional force is called static force. It is possible to determine how much tilt the robot is experiencing by measuring the gravitational force. In this measurement, robots are able to balance themselves, or determine whether they are driving flat or uphill. Dynamic force can be measured using an accelerometer to determine robot velocity and speed. (4) Temperature sensor ( 4): Temperature sensors measure the temperature change around them.
According to theory, a change in voltage will provide the equivalent temperature value of a change in temperature. The following are some commonly used temperature sensor ICs: LM34, LM35, TMP35, TMP37 etc. (5) Sound sensor ( 5): It is generally a microphone that detects sound and returns a voltage corresponding to the sound level. Using sound sensors, a simple robot can navigate by listening to sounds. Unlike light sensors, sound sensors generate a very small voltage difference which will be amplified to produce measurable voltage changes.
A set of selection criteria for sensors is given. In order to compare the sensors, you should consider the following factors: (1) Resolution (e 1 ): The smallest change that can be detected by the sensor. There is no guarantee that high is always positive. It would pick up even very minute fluctuations, requiring additional processing if it is too high. (2) Sensitivity (e 2 ): An indicator of the ratio of change in output to change in input. Sensitivity that is too high can be problematic. It is also true that in most cases, the higher the sensitivities are the higher the costs. Step 1: Table 1 is created using the DM data. Step 2: A normalized decision matrix is shown in Table 2.  Step 4: Both ideal values are listed as follows: Step 5: To calculate the HD between each option with ideal values, we use the values in Tables 4  and 5.    Step 6: We calculate each option's relative closeness value (see Table 6).  Step 7: The ranking of alternatives is 5 2 4 3 1 . Consequently, selecting the most suitable sound sensor 5 is a great solution.

Comparison of suggested and existing models
In this subsection, we give a comparison between the proposed models and some of the existing models, demonstrating their applicability and advantages. We used Type-II FNWA, Type-II FNWG, Type-II GFNWA and Type-II GFNWG approaches based on HD. The different distances are shown in Table 7. On the basis of these four aggregating operators, the HDs of proposed strategies and existing techniques are shown in Figure 1. By altering various values of δ, we can obtain Table 8.  Figure 2. The TOPSIS model in Figure 2 is based on a Type-II FNWA technique.
A graph of the different values for the Type-II FNWA operator is presented in Figure 3.

Sensitivity analysis
The aforementioned analysis demonstrates that, the ranking of the alternatives is 5 4 2 3 1 when 2 δ 12. In the event when 13 δ 33, the ordering of the alternatives becomes when δ = 34. As a result, the preferred option was changed from 5 to 4 . The basis of the alternate rankings is similarly composed of Type-II FNWG, Type-II GFNWA and Type-II GFNWG operators with values of δ.

Advantages
This paper proposes a method that offers the following benefits, as shown by the above analysis. In the article, NFN and neutrosophic number are combined to introduce the concept of Type-II FNN. A Type-II FNN interprets human behavior and natural phenomena that have a normal distribution in nature and describes ambiguous information. Therefore, type-II FNN information is characterized more broadly and is more in line with how humans think. The decision-maker may use parameter δ to select the decision result according to their preferences. In this paper, the mode of ranking alternatives will be discussed in terms of Type-II FNWAs, Type-II FNWGs, Type-II GFNWAs and Type-II GFNWGs.

Conclusions
We used HD for Type-II FNNs, whose algebraic accessibility provides an added benefit. The HD of Type-II Fermatean sets has a wide range of practical applications in data analysis. Using persuasive statistical examples, the benefits of the HD are demonstrated. Many illustrations of Type-II FNWA, Type-II FNWG, Type-II GFNWA and Type-II GFNWG with AOs were shown, and for the first time, models were proposed. The Type-II FNS method can assist persons in selecting the best alternative from a selection of choices. We have proposed changes for the Type-II FNWA, Type-II FNWG, Type-II GFNWA, and Type-II GFNWG AOs to the MADM problem on δ. The distinct ranking of options may be found for the Type-II FNWA, Type-II FNWG, Type-II GFNWA, and Type-II GFNWG AOs on δ. The results demonstrated that the generalized values of δ have the most impact on the ranking of options at the last stage. The DM may set the values of δ in accordance with the present parameters and then proceed to make appropriate decisions in order to attain the best reasonable ranking. As a result, the values of δ can provide guidance to the decision-maker in arriving at the outcome. Since this is a wide open field, the ideas presented here will be useful to future academics with an interest in the subject. Ideas require a great deal of work because of their applicability to real-world issues. Graphs of Fermatean fuzzy sets and aggregation operators on these sets are some directions that will help us address a variety of real-world problems.