Multi-criteria decision-making algorithm based on aggregation operators under the complex interval-valued q-rung orthopair uncertain linguistic information
Available accessResearch articleFirst published online August 11, 2021
Multi-criteria decision-making algorithm based on aggregation operators under the complex interval-valued q-rung orthopair uncertain linguistic information
The paper aims to present a concept of a Complex interval-valued q-rung orthopair uncertain linguistic set (CIVQROULS) and investigated their properties. In the presented set, the membership grades are considered in terms of the interval numbers under the complex domain while the linguistic features are added to address the uncertainties in the data. To further discuss more, we have presented the operation laws and score function for CIVQROULS. In addition to them, we present some averaging and geometric operators to aggregate the different pairs of the CIVQROULS. Some fundamental properties of the proposed operators are stated. Afterward, an algorithm for solving the decision-making problems is addressed based on the proposed operator using the CIVQROULS features. The applicability of the algorithm is demonstrated through a case study related to brain tumors and their effectiveness is compared with the existing studies.
Cancer is the uncontrolled advancement of bizarre cells wherever in the body. These atypical cells are named illness cells, undermining cells, or tumor cells. These cells can attack run-of-the-mill body tissues. Various dangerous developments and the strange cells that make the sickness tissue are furthermore recognized by the name of the tissue that the sporadic cells began from (for example, chest infection, cell breakdown in the lungs, colorectal illness). Harm is not limited to individuals; animals and other living creatures can get sickness. Coming up next is a schematic that shows the common cell division and how when a cell is hurt or changed without a fix to its system, the cell typically fails horrendously. In a like manner exhibited is what happens when such hurt or unrepaired cells do not fail miserably and become infection cells and show uncontrolled division and advancement - a mass of threat cells make. Consistently, sickness cells can part away from this exceptional mass of cells, travel through the blood and lymph structures, and in various organs where they can again repeat the uncontrolled advancement cycle. This pattern of threat cells leaving a domain and filling in another body district is named metastatic spread or metastasis. For example, if chest danger cells spread to a bone, it infers that the individual has a metastatic chest infection to the bone. This is not comparable to “bone infection,” which would mean the threat had started in the bone. Firstly, the theory of cancer was explored by Black [1] in 1991. Moreover, numerous scholars have utilized the theory of cancer in the environment of FS [2].
Decision making procedure is a capable tool to handle awkward and intricated information in daily life issues. In the last few years, numerous scholars have implemented the decision-making technique in the environment of separated areas. In the genuine decision-making process, a significant issue is a way to show the quality worth even more effectively and precisely. Because of the multifaceted nature of decision-making issues and the fuzziness of decision-making conditions, it is not enough to show attribute objects of alternatives by objects. To survive through such forms of concerns, the theory of intuitionistic fuzzy set (IFS) was investigated by Atanassove [3], by including the degree of falsity in the idea of the fuzzy set (FS) which was developed by Zadeh [4]. The rule of IFS is that the sum of both degrees’ is cannot be exceeded from the unit interval, have extensive effective and superior to FS to deal with those issues which cannot be resolved by using FS theory. Additionally, the theory of interval-valued IFS (IVIFS) was diagnosed by Atanassove [5], whose both degrees are in the form of the subset of the unit interval. IVIFS is the modified version of the interval-valued FS (IVFS) to handle difficult information. Several scholars have implemented the theory of IFS in numerous areas [6–9]. Moreover, the theory of Pythagorean fuzzy set (PFS) was investigated by Yager [10], by using the rule that is the sum of the square of both degrees’ is cannot be exceeded from the unit interval, have extensive effective and superior to IFS to deal with those issues which cannot be resolved by using IFS theory. Additionally, the theory of interval-valued PFS (IVPFS) was diagnosed by Garg [11], whose both degrees are in the form of the subset of the unit interval. IVPFS is extensively reliable to handle difficult information. Several scholars have implemented the theory of PFS in numerous areas [12–16]. But there were numerous worries if a decision-maker gives such types of issues, whose sum of the squares of both degrees’ cannot be exceeded from unit interval. Moreover, the theory of q-rung orthopair fuzzy set (QROFS) was investigated by Yager [17], by using the rule that is the sum of the q-power of both degrees’ cannot be exceeded from the unit interval, have extensive effectiveness, and superior to PFS to deal with those issues which cannot be resolved by using PFS theory. Additionally, the theory of interval-valued QROFS (IVQROFS) was diagnosed by Joshi et al. [18], whose both degrees are in the form of the subset of the unit interval. IVQROFS is extensive reliable to handle difficult information. Several scholars have implemented the theory of QROFS in numerous areas [19–21].
Complex IFS (CIFS) was investigated by Alkouri and Salleh [22], by including the degree of falsity in the idea of complex FS (CFS) which was developed by Ramot [23]. CIFS contains the degree of truth and falsity in the form of a complex number with a rule that is the sum of the real part (also for an imaginary part) of both degrees’ is cannot be exceeded from the unit interval, have extensive effective and superior to CFS to deal with those issues which cannot be resolved by using CFS theory. Additionally, the theory of interval-valued CIFS (IVCIFS) was diagnosed by Garg [24], whose both degrees are in the form of the subset of the unit interval. IVCIFS is the modified version of the interval-valued CFS (IVCFS) to handle difficult information. Several scholars have implemented the theory of CIFS in numerous areas [25–28]. Moreover, the theory of complex PFS (CPFS) was investigated by Ullah et al. [29], by using the rule that is the sum of the square of the real part (also for an imaginary part) of both degrees’ is cannot be exceeded from the unit interval, have extensive effective and superior to CIFS to deal with those issues which cannot be resolved by using CIFS theory. Several scholars have implemented the theory of CPFS in numerous areas [30]. But there were numerous worries if a decision-maker gives such types of issues, whose sum of the squares of the real part (also for an imaginary part) of both degrees’ is cannot be exceeded from unit interval. Moreover, the theory of complex QROFS (CQROFS) was investigated by Liu et al. [31, 32], by using the rule that is the sum of the q-power of the real part (also for an imaginary part) of both degrees’ is cannot be exceeded from the unit interval, have extensive effective and superior to CPFS to deal with those issues which cannot be resolved by using CPFS theory. Additionally, the theory of complex IVQROFS (CIVQROFS) was diagnosed by Garg et al. [33], whose both degrees are in the form of the subset of the unit interval. CIVQROFS is extensive reliable to handle difficult information. Several scholars have implemented the theory of CQROFS in numerous areas [34–40].
In numerous genuine issues, it is difficult for decision-makers to give their evaluations in quantitative articulations. For instance, when a specialist assesses an up-and-comer’s degree of proficient capability, he may feel increasingly helpful or progressively acquainted with utilizing semantic terms, for example, “excellent”, “great”, or “medium”, to communicate their judgment. The theory of linguistic variables (LVs) was presented by Zadeh [41]. Xu [42] discovered the uncertain linguistic sets (ULS). Liu and Jin [43] explored some aggregation operators based on intuitionistic fuzzy uncertain linguistic sets, and Lu and Wei [44] considered Pythagorean uncertain linguistic aggregation operators. Liu et al. [45] presented the q-rung orthopair uncertain linguistic aggregation operators and their application in the MADM problem. Gao and Wei [46] presented the interval-valued Pythagorean uncertain linguistic aggregation operators, Kan et al. [47] discovered the theory of interval-valued intuitionistic uncertain linguistic aggregation operators.
CIFSs, CIVIFSs, CPFSs, CIVPFSs, CQROFSs, and CIVQROFSs theory have been widely used by the researchers, but due to the complexity of the decision-making problems, sometimes decision-makers are not suitable to provide their judgment in form of single-valued membership, non-membership degrees, and uncertain linguistic terms in the form of interval-valued with a rule that is the sum of the real part of the membership grades are restricted with united interval. Consequently, an extension of the existing theories might be extremely valuable to depict the uncertainties because of his/her reluctant judgment in complex decision-making problems. Therefore, to provide more freedom to the decision-makers, it is advisable to ask the experts to describe their preferences employing intervals.
The same situation occurs in CIVIFULSs and CIVPFULSs when the decision-maker provides these types of data, which do not satisfy the conditions of CIVIFULSs and the condition of CIVPFULSs. For example, a decision-maker provides [0.4, 0.6] ei2π[0.3,0.4] for interval-valued membership grade and [0.3, 0.4] ei2π[0.2,0.3] for interval-valued non-membership grade and assign to uncertain linguistic terms, the CIVIFULSs and CIVPFULSs cannot describe this result. For dealing with such types of situations, in this article, we examine the novel approach of complex interval-valued q-rung orthopair fuzzy uncertain linguistic sets (CICQROFULSs) and their fundamental operational laws. Their eminent characteristic is that the sum of the qth power of the real part (Similarly for an imaginary part) of interval-valued membership degree and the qth power of the real part (Similarly for an imaginary part) of interval-valued non-membership degree is equal to or less than 1. The proposed CIVQROFULSs is an important technique to deal with uncertain and more difficult information and then apply it to solve the multi-attribute decision-making (MADM) problems. The CIVQROFULSs are more generalized than existing methods like complex interval-valued Pythagorean fuzzy uncertain linguistic set (CIVPFULS) and complex interval-valued intuitionistic fuzzy uncertain linguistic set (CIVIFULS). The advantages of the elaborated ideas are discussed are below:
If we will take the imaginary part as zero, in the terms of membership grade and non-membership grade, then the proposed approach is to convert it into an interval-valued q-rung Orthopair fuzzy uncertain linguistic set (IVQROFULS). IVQROFULS is the special case of the proposed method.
If we will take the imaginary part as zero, in the terms of membership grade and non-membership grade with taking the value of q = 1, then the proposed approach is to convert into the interval-valued intuitionistic fuzzy uncertain linguistic set (IVIFULS).
If we will take the imaginary part as zero, in the terms of membership grade and non-membership grade with taking the value of q = 2, then the proposed approach is to convert into an interval-valued Pythagorean fuzzy uncertain linguistic set (IVPFULS).
If we considered the value of parameter q = 1 in the environment of CIVQROFULS, then the CIVQROFULS is converted for the complex interval-valued intuitionistic fuzzy uncertain linguistic set (CIVIFULS).
If we considered the value of parameter q = 2 in the environment of CIVQROFULS, then the CIVQROFULS is converted for the complex interval-valued Pythagorean fuzzy uncertain linguistic set (CIVPFULS).
To broaden the scope of supporting grade reaching out from the unit plate in the form of a complex number belonging to the unit disc in a complex plane, the summary of the investigated approaches is discussed below.
To explore the CIVQROULS and their properties.
To investigate CIVQROULWA and CIVQROULWG operators are investigated.
Some cases of the presented approaches are also discussed.
To review the types of brain tumors. By using the investigated operators based on CIVQROULSs, the dangerous types of brain tumors are examined.
To find the validity and proficiency of the explored works, we resolved some numerical examples by using the proposed operators.
The advantages, comparative analysis, and graphical expressions of the discovered theory are also discussed. The elaborated approaches are also discussed with the help of Fig. 1.
Geometrical expressions of the elaborated approaches.
The rest of the manuscript is organized in the following ways: In section 2, we recall the idea of CIVQROFSs and its laws. In section 3, we developed the theory of CIVQROULS and its properties. In section 4, the idea of aggregation operators is utilized in the environment if CIVQROULS are called CIVQROULWA and CIVQROULWG operators are investigated. Some cases of the presented approaches are also discussed. In section 5, we reviewed the types of brain tumors, which are the uncontrolled development of anomalous cells in the body. By using the investigated operators based on CIVQROULSs, the dangerous types of brain tumors are examined. To find the validity and proficiency of the explored works, we resolved some numerical examples by using the proposed operators. Finally, the advantages, comparative analysis, and graphical expressions of the discovered theory are also discussed. The conclusion of this manuscript is discussed in section 6.
Preliminaries
In this study, we recall some existing ideas like CIVQROFS and its laws. The idea of LVs and ULSs are their related laws are also studied. Moreover, the terms XU, μ, and η shown the universal, the grade of truth, and the grade of falsity with qSC, δSC ⩾ 1. The theory of complex q-rung orthopair fuzzy set (CQROFS) was investigated by Liu et al. [31, 32], by using the rule that is the sum of the q-power of the real parts of both degrees cannot be exceeded from the unit interval, have extensive effectiveness, and superior to CPFS to deal with those issues which cannot be resolved by using CPFS theory. Additionally, the theory of complex interval valued QROFS (CIVQROFS) was diagnosed by Garg et al. [33], whose both degrees are in the form of the subset of the unit interval.
where and , with conditions: . Moreover, is called refusal grade, the complex interval-valued q-rung orthopair fuzzy number (CIVQROFN) is represented by
Definition 2: [33] For any two CIVQROFNs and , then
Definition 3: [33] For any two CIVQROFNs and , the score and accuracy function is given by:
By using Equations (2) and (3), we demonstrated the following rules:
If , the ;
If , then:
If , the ;
If , the .
In numerous genuine issues, it is difficult for decision-makers to give their evaluations in quantitative articulations. For instance, when a specialist assesses an up-and-comer’s degree of proficient capability, he may feel increasingly helpful or progressively acquainted with utilizing semantic terms, for example, “excellent", “great", or “medium", to communicate their judgment. The theory of linguistic variables (LVs) was presented by Zadeh [41]. Xu [42] discovered the uncertain linguistic sets (ULS).
Definition 4: [41] The set with odd cardinality is called linguistic term set, where, ʑ is the cardinality of , and is a linguistic variable. A possible linguistic term set is given by:
by using the values of ʑ = 7, then ʑ - 1 =6. The linguistic terms are expressed by Pythagorean fuzzy sets for five and seven terms. Further, is called continuous linguistic term sets if the following conditions hold:
The ordered set: iff θ < φ;
The negation operator: such that φ = 2 - θ;
if θ ⩽ φ, then and .
Definition 5: [42] For an ULSs is the upper and lower limits of with 0 < θ ⩽ φ. For any two ULVs and , then
;
;
;
.
Complex interval valued q-rung orthopair uncertain linguistic variables
In this study, the theory of CIVQROULS and its related laws are investigated. The CIVQROULS is an extensive useful technique to cope at a time with four dimensions information’s in real-life issues. The CIVQROULS is demonstrated below.
Definition 6: A CIVQROULS is demonstrated by
where and , with conditions: with a ULV . Moreover, is called refusal grade, the complex interval-valued q-rung orthopair uncertain linguistic number (CIVQROULN) or complex interval-valued q-rung orthopair uncertain linguistic variable (CIVQROULV) is represented by .
Definition 7: For any two CIVQROULNs and , then
Where ; , , , ; ; .
Where , , , ; ;
Where ; , , , ; ;
Where ; , , , ; ;
Where ; g (x, y) = xy; ; g (x) = (x) δSC for any (x, y) ∈ [0, 1] × [0, 1].
Definition 8: For any two CIVQROULNs
and , the expectation and accuracy function is given by:
Based on the above two notions, the compassion between two CIVQROULNs is given by:
If , the ;
If , then:
If , the ;
If , the .
Theorem 1:For any two CIVQROULNs and with scalers δSC-1, δSC-2 ⩾ 0, then
;
;
;
;
;
.
Proof: Straightforward.
Aggregation operators for CIVQROULSs
In this study, we developed the idea of CIVQROULWA, CIVQROULWG operators and studied their properties. Some cases of the proposed work are also discussed.
Definition 9: For a collection CIVQROULNs , the CIVQROULWA operator is given by:
where ωw = (ωw-1, ωw-2, …, ωw-n) T denotes the weight vectors with a condition .
Theorem 2: Suppose a collection CIVQROULNs , the aggregated value of the Equation (7) is again a CIVQROULN, we have
Proof: By following the operation laws, we can easily derive that. Hence, we omit the proof here.
Further, we evaluate some properties for CIVQROULNs like idempotency, monotonicity, and boundedness.
Theorem 3:Suppose a collection CIVQROULNs , j = 1, 2, …, nand , if , then
Proof: Suppose , then by using Equation (8), we have
The result has been proved.
Theorem 4:Suppose a collection CIVQROULNs and , if and , then
First, we have study about the uncertain linguistic parts and , we have
Hence , similarly, we can prove that . Next, we discuss the real part of the complex-valued truth grade , we have
The upper and imaginary part of the complex-valued truth grade is the same. Next, we prove the real part of the complex-valued falsity grade , we have
The upper complex-valued falsity grade is the same. Hence the expectation values of the and then by using the Def. (8), we have If , the . If , then: If , the because and . So, by using the above properties, we have the result, such that
The result has been proved.
Theorem 5:Suppose a collection CIVQROULNs , if and , then
Proof: It is clear that
and and , then by using theorem 3 and theorem 4, such that
therefore
The result has been proved.
Remark 1: If we choose the values of the imaginary part as zero in Equation (8), then the Equation (8) is reduced for interval-valued q-rung orthopair uncertain linguistic sets. Similarly, if we choose the values of qSC = 2 in Equation (8), then the Equation (8) is reduced for complex interval-valued Pythagorean uncertain linguistic sets and if we choose the values of qSC = 1 in Equation (8), then the Equation (8) is reduced for complex interval-valued intuitionistic uncertain linguistic sets.
Definition 10: For a collection CIVQROULNs , the CIVQROULWG operator is given by:
where ωw = (ωw-1, ωw-2, …, ωw-n) Tdenotes the weight vectors with a condition .
Theorem 6: Suppose a collection CIVQROULNs , the aggregated value of the Equation (9) is again a CIVQROULN, we have
Proof: Straightforward. (Similar to theorem 2).
Further, we evaluate some properties for CIVQROULNs like idempotency, monotonicity, and boundedness.
Theorem 7:Suppose a collection CIVQROULNs and , if , then
Proof: Straightforward. (Similar to theorem 3).
Theorem 8:Suppose a collection CIVQROULNs and , if and , then
Proof: Straightforward. (Similar to theorem 4).
Theorem 9:Suppose a collection CIVQROULNs , if and , then
Proof: Straightforward. (Similar to theorem 5).
Remark 2: If we choose the values of the imaginary part as zero in Equation (10), then the Equation (10) is reduced for interval-valued q-rung orthopair uncertain linguistic sets. Similarly, if we choose the values of qSC = 2 in Equation (10), then the Equation (10) is reduced for complex interval-valued Pythagorean uncertain linguistic sets and if we choose the values of qSC = 1 in Equation (10), then the Equation (10) is reduced for complex interval-valued intuitionistic uncertain linguistic sets.
Classification of brain tumors by using investigated approaches
Basic Brain tumors start in the cerebrum and occur as the result of irregular changes to neural connections, known as changes. As the cells change, they create and copy uncontrollably, forming a mass, or tumor. Tumors that have spread (metastasized) to the brain from various zones in the body are known as cerebrum metastases. Cerebrum metastases can be a single tumor or various tumors.
Brain tumors have more than 120 sorts, according to the National Brain Tumor Society. Some cerebrum tumors, for instance, a glioblastoma multiforme, are perilous and may be rapidly creating. Various kinds of brain tumors, for instance, a meningioma, maybe moderate creating and liberal. Basic psyche tumors structure in neurotransmitters and are requested by the sort of cell or where in the cerebrum they at first make. For instance, astrocytoma’s structure in star-shaped cells is called astrocytes. Pituitary tumors are found in the pituitary organ at the lower part of the cerebrum. The most notable basic cerebrum tumors are called gliomas, which start in the glial (solid) tissue. Around 33% of all fundamental cerebrum tumors and other tangible framework tumors structure from glial cells.
Many brain tumors are cancer-causing. For example, most of all gliomas examined in adults are glioblastomas, a powerful sort of cerebrum infection. Ependymomas and oligodendrogliomas in like manner are kinds of cerebrum tumors that may be unsafe. However, not all cerebrum tumors are hazardous. Various meningiomas, craniopharyngiomas, and pituitary tumors are sympathetic. That is the explanation it is fundamental to get a cautious and definite investigation of a cerebrum tumor. It is similarly basic to appreciate that even compassionate tumors can hurt cerebrum tissue and cause results, for instance, headaches, shortcoming, and twofold or clouded vision. So, whether a cerebrum tumor is not ruinous, tolerating fortunate and legitimate treatment may be essential to your overall prosperity. But here we discuss five types of brain tumors which are discussed below.
Types of Brain Cancer
Astrocytoma’s Ǥ
Astrocytoma’s which are the most broadly perceived CNS tumor, arise wherever in the cerebrum or spinal line and make from pretty much nothing, star-formed cells called astrocytes. In adults, astrocytoma’s often occurred in the cerebrum, the greatest bit of the psyche. The cerebrum uses unmistakable information to refer us to what is going on around us and how the body should respond. The cerebrum moreover controls talk, improvement, and emotions, similarly as scrutinizing, thinking, and learning. Brain stem gliomas are such an astrocytoma that structures in the cerebrum stem, which controls various fundamental limits, for instance, interior warmth level, circulatory strain, breathing, craving, and thirst. The cerebrum stem similarly sends all the signs to the body from the psyche. The psyche stem is in the most negligible bit of the cerebrum and interfaces the cerebrum and spinal rope. Tumors here can be difficult to treat. Most cerebrum stem gliomas are high-grade astrocytoma’s, which is explained more clearly with the help of Fig. 2.
Expressed the Astrocytoma’s.
Glioblastoma multiforme
Glioblastoma multiforme, in any case, called glioblastoma, GBM, or assessment IV astrocytoma, is a rapidly creating intense kind of CNS tumor that structures on the solid tissue of the psyche. Glioblastoma is the most notable assessment IV brain harm. Glioblastomas may appear in any projection of the psyche, anyway, they develop even more typically in the frontal and brief folds. Glioblastomas conventionally impact adults, which is explained more clearly with the help of Fig. 3.
Expressions of Glioblastoma Multiforme.
Meningioma
Meningioma makes in the cells of the layer that include the psyche and spinal line. Meningiomas (also called meningeal tumors) speak to around 15 percent of each intracranial tumor. By far most of these tumors are generous (non-risky and moderate creating). Meningiomas are regularly taken out with an operation. A couple of meningiomas may not need brief treatment and may remain undetected for a seriously long time. Most meningiomas are dissected in women someplace in the scope of 30 and 50 years old. For simplicity, we explained meningioma cancer with the help of Fig. 4.
Expressed the Meningioma.
Craniopharyngiomas
Craniopharyngiomas make in the zone of the cerebrum near the pituitary organ (the guideline endocrine organ that produces hormones that control various organs and many-body limits, especially improvement) near the operational hub. These cerebrum tumors are by and large generous. In any case, they may from time to time be hazardous considering the way that they may make strain on, or hurt, the operational hub and impact basic capacities, (with regards to the model, interior warmth level, longing, and thirst). These tumors happen much of the time in adolescents and young people, or adults over age 50. For simplicity, we explained the craniopharyngiomas cancer with the help of Fig. 5.
Expressed the Craniopharyngiomas.
Germ cell tumors
Germ cell tumors arise out of having sex (egg or sperm) cells, in any case, called germ cells. The most notable kind of germ cell tumor in the cerebrum is germinoma. Next to the brain, germinomas can outline in the ovaries, balls, chest, and waist. Most germ cell tumors occur in youths. For simplicity, we explained the germ cell tumors cancer with the help of Fig. 6.
Expressed the Germ Cell Tumors.
To examine which kind of brain cancer are dangerous for our human beings, we consider the family of alternatives and their attributes with weight vectors, whose expressions are summarized are following as: and °F1, °F2, °F3, …, °Fm with weight vectors like Θ = (Θ1, Θ2, . . , ΘΛ) T represented the weight vectors with . The attributes are basically expressed the symptoms of the brain tumors which are initiated by: °F1: Drowsiness, °F2: Impaired Speech, °F3: Seizures, and °F4: Difficulty in swallowing. For this, we construct the decision matrix which is denoted and defined by: and their weight vectors such that Θ = (0.3, 0.3, 0.3, 0.1) T, the steps of the algorithm are summarized are follow as:
Based on the CIVQROFNs, we construct the decision matrix, whose very items in the form of CIVQROFN.
Based on the CIVQROULWA and CIVQROULWG operators, we aggregate the decision matrix, which is given in step 1.
Additionally, by using the score function, we examine the Score values of the aggregated values in step 2.
Based on the Score values, we examine the ranking values for proposed works are examine the best options from the family of alternatives.
For this, based on the CIVQROFNs, we construct the decision matrix, whose very items in the form of CIVQROFN, which is discussed in the form of Table 1.
Original decision matrix
°F1
°F2
°F3
°F4
Based on the CIVQROFNs, we construct the decision matrix, whose very items in the form of CIVQROFN, which is discussed in the form of Table 1. Based on the CIVQROULWA and CIVQROULWG operators, we aggregate the decision matrix, which is given in the form of Table 1 and the aggregated values are discussed in the form of Table 2.
Expresses the aggregated values by using CIVQROULWA and CIVQROULWG
Alternatives
CIVQROULWA
CIVQROULWG
Additionally, by using the score function, we examine the Score values of the aggregated values in Table 2 are discussed in Table 3. The geometrical representation of the investigated score values of Table 3 is discussed in the form of Fig. 7.
Geometrical expressions of the information in Table 3.
Based on the Score values, we examine the ranking values for proposed works are examine the best options from the family of alternatives as .
The explored operators like CIVQROULWA and CIVQROULWG operators gives the results that the best alternatives are , which shows the germ cell tumors. The graphical representation of Table 3 is shown in Fig. 7. Further, to check the reliability and consistency of the investigated operators, we have chosen complex interval valued pythagorean uncertain linguistic information (CIVPULI) and resolved it by using developed operators. The CIVPULIs are discussed in the form of Table 4.
Original decision matrix, which contains the CIVPULIs
°F1
°F2
°F3
°F4
Based on the CIVQROFNs, we construct the decision matrix, whose very items in the form of CIVQROFN, which is discussed in the form of Table 4. Based on the CIVQROULWA and CIVQROULWG operators, we aggregate the decision matrix, which is given in the form of Table 4 and the aggregated values are discussed in the form of Table 5.
Expresses the aggregated values by using CIVQROULWA and CIVQROULWG
Alternatives
CIVQROULWA
CIVQROULWG
Additionally, by using the score function, we examine the Score values of the aggregated values in Table 5 are discussed in Table 6. The geometrical representation of the investigated score values of Table 6 is discussed in the form of Fig. 8.
Geometrical expressions of the information in Table 6.
Based on the Score values, we examine the ranking values for proposed works are examine the best options from the family of alternatives.
The explored operators like CIVQROULWA and CIVQROULWG operators give the results that the best alternatives are , which shows the germ cell tumors and their graphical representation is shown in Fig. 8. Further, to check the reliability and consistency of the investigated operators, we have chosen complex interval-valued q-rung orthopair uncertain linguistic information (CIVQROULI) and resolved it by using developed operators. The CIVQROULIs are discussed in the form of Table 7.
Original decision matrix, which contains the CIVQROULIs
1
2
3
4
Based on the CIVQROFNs, we construct the decision matrix, whose very items in the form of CIVQROFN, which is discussed in the form of Table 7. Based on the CIVQROULWA and CIVQROULWG operators, we aggregate the decision matrix, which is given in the form of Table 7 and the aggregated values are discussed in the form of Table 8.
Expresses the aggregated values by using CIVQROULWA and CIVQROULWG
Alternatives
CIVQROULWA
CIVQROULWG
Additionally, by using the score function, we examine the Score values of the aggregated values in Table 8 are discussed in Table 9.
Based on the Score values, we examine the ranking values for proposed works are examine the best options from the family of alternatives as . The explored operators like CIVQROULWA and CIVQROULWG operators give the results that the best alternatives are , which shows the Meningioma tumors. The geometrical representation of the investigated score values of Table 9 is discussed in the form of Fig. 9.
Geometrical expressions of the information in Table 9.
Comparative analysis
Moreover, to find the validity and proficiency of the discovered operators with the help of existing operators based on interval-valued q-rung orthopair uncertain linguistic information’s, interval-valued Pythagorean uncertain linguistic information’s, and interval-valued intuitionistic uncertain linguistic information’s, who is detailed are discussed are follow as: CIVQROULWA, CIVQROULWG, complex interval-valued Pythagorean uncertain linguistic weighted averaging (CIVPULWA), complex interval-valued Pythagorean uncertain linguistic weighted geometric (CIVPULWG), complex interval-valued intuitionistic uncertain linguistic weighted averaging (CIVIULWA), complex interval-valued intuitionistic uncertain linguistic weighted geometric (CIVIULWG), interval-valued q-rung orthopair uncertain linguistic weighted averaging (IVQROULWA), interval-valued q-rung orthopair uncertain linguistic weighted geometric (IVQROULWG), Gao and Wei [46] presented the interval-valued Pythagorean uncertain linguistic aggregation operators, Kan et al. [47] discovered the theory of interval-valued intuitionistic uncertain linguistic aggregation operators. Based on the information in Table 1 the comparative analysis is discussed in the form of Table 10.
From Table 10, we can get (1) CIVIFULS cannot express the information described by CIVQROFULS; (2) CIVPFULS cannot express the information described by CIVQROFULS; (3) the proposed method in this paper can the same ranking results and the methods in [46, 47], which cannot show the effectiveness of the proposed method because the CICQROFULS is reduced into CIVPFULS when q = 2 and the CICQROFULS is reduced into CIVIFULS when q = 1.
The geometrical representation of the investigated score values of Table 10 is discussed in the form of Fig. 10. The geometrical representation of the investigated score values of Table 10 is discussed in the form of Fig. 10. The geometrical representation of the investigated score values of Table 11 is discussed in the form of Fig. 11. The geometrical representation of the investigated score values of Table 12 is discussed in the form of Fig. 12.
Comparative analysis, by using the information’s of Table 1
Similarity-based on the information of Table 4, the comparative analysis is discussed in the form of Table 11. Similarity-based on the information of Table 7, the comparative analysis is discussed in the form of Table 12.
From Table 11, we can get (1) CIVIFULS cannot express the information described by CIVQROFULS; (2) CIVPFULS can express the information described by CIVQROFULS; (3) the proposed method in this paper can the same ranking results and the methods in [46, 47], which cannot show the effectiveness of the proposed method because the CICQROFULS is reduced into CIVPFULS when q = 2 and the CICQROFULS is reduced into CIVIFULS when q = 1.
From Table 12, we can get (1) CIVIFULS can express the information described by CIVQROFULS; (2) CIVPFULS can express the information described by CIVQROFULS; (3) the proposed method in this paper can the same ranking results and the methods in [46, 47], which cannot show the effectiveness of the proposed method because the CICQROFULS is reduced into CIVPFULS when q = 2 and the CICQROFULS is reduced into CIVIFULS when q = 1. For simplicity, we have drawn the figures and the Figs. 9 to 11 contains six sub-figures that shown the stability of the existing and explored operators. Figure 6 contains six sub-figures, and the last figure is represented the proposed operators for q = 8, and the other existing operators cannot be able to cope with it. Therefore, the investigated operators based on CIVQROULNs are extensive reliable, and more valid is compare to other existing drawbacks [41–47].
Conclusion
The tumor is the uncontrolled development of anomalous cells in the body. Tumor creates when the body’s ordinary control system quits working. Old cells do not bite the dust and rather outgrow control, framing new, irregular cells. These additional phones may frame a mass of tissue, called a tumor. A few malignant growths, for example, leukemia, do not shape tumors. In this manuscript, firstly, we developed the theory of CIVQROULS and its properties. Moreover, the idea of aggregation operators is utilized in the environment if CIVQROULS are called CIVQROULWA and CIVQROULWG operators are investigated. Some cases of the presented approaches are also discussed. Secondly, we reviewed the types of brain tumors. By using the investigated operators based on CIVQROULSs, the dangerous types of brain tumors are examined. To find the validity and proficiency of the explored works, we resolved some numerical examples by using the proposed operators. Finally, the advantages, comparative analysis, and graphical expressions of the discovered theory are also discussed. Some future study may involve associating such Dombi AOs with power operations and Heronian mean operators under the different fuzzy environment [48–51].
Footnotes
Acknowledgments
Authors are grateful for the financial help provided by the Beijing Social Science Foundation of China (No. 18GLC082) and the Taif University Researchers Supporting project number (TURSP-2020/73), Taif University, Taif, Saudi Arabia
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