#### Abstract

With the frequent occurrence of emergency events, decision-making (DM) plays an increasingly significant role in coping with them and has become an important and the challenging research focus recently. It is critical for decision makers to make accurate and reasonable emergency judgments in a short period as poor decisions can result in enormous economic losses and an unstable social order. As a consequence, this work offers a new DM approach based on novel distance and similarity measures using *q*-rung linear Diophantine fuzzy (*q*-RLDF) information to assure that DM problems may be addressed successfully and fast. One of the useful methods for determining the degree of similarity between the objects is the similarity measure. In this paper, we propose some new *q*-rung linear Diophantine fuzzy (*q*-ROLDF) distances and similarity measures. The Jaccard similarity measure, exponential similarity measure, and cosine and cotangent function-based similarity measures are proposed for *q*-LDFSs. The defined similarity measures are applied to the logistics and supply chain management problem, and the results are discussed. A comparison of new similarity measures is developed, and the proposed work’s advantages are discussed.

#### 1. Introduction

Models of knowledge-based decision-making attract substantial attention in industry and academia. A significant number of original research and thesis studies were undertaken to create effective decision support systems (DSSs) to promote managerial decision. DSS is classified as a specific class of electronic data system which supports decision-making (DM) management activities. In the early 1970s, the DSS idea was developed by Scott Morton’s work. The methodology aims to evaluate strategic decisions in a complex and poorly organized environment, providing guidance to decision makers. DSSs have some huge benefits in decision-making by helping decision makers in their challenges and improve the effectiveness of the decision-making process [1].

For the purpose of logistics support, the better selection of logistic providers (LPs) is now important with the growth of the supply chain theories. Over the past decades, there has been a drastic change in the course of DSS [2]. Computer and technical experts have made great efforts to systematic processes of decision-making in the engineering and manufacturing industries [3]. Zha et al. [4], for example, developed a compromise decision-supporting issue approach and the fuzzy synthetic decision model (FSDM). Implementation of the DSS of multicriteria decision-making (MCDM) in supply chains is a constant challenge [5–7]. The reliability of the decision criteria in many decision-making problems (DMPs) is strictly dependent as external weights on the stakeholders and customer expectations. The delivery of decisions through solid and numerical values is difficult for clients. Fuzzy linguistic variables enable us to ensure judgmental consistency.

Zadeh [8] developed the concept of fuzzy set (FS) in 1965, which addressed the membership degree (MD), and Atanassov [9] generalized the concept of FS by including nonmembership degree (NMD). IFS is a significant generalization of fuzzy sets that has been developed as a useful tool for understanding uncertain data. Pattern detection, decision-making, cluster analysis, and a variety of other areas are all derived from the IFS. In IFS theory, an element’s MD and NMD must be between 0 and 1, and the number of MD and NMD must be between 0 and 1. As a result, in some practical applications, the values of MD and NMD of an element whose number of squares of MD and NMD may be greater than one are assigned to decision makers or decision experts. In such instances, the IFS has failed to clarify true or uncertain facts. Yager addressed this form of vague data by adding a new concept Pythagorean fuzzy set (PFS). Yager increased the space of MD and NMD by requiring that the square sum of MD and NMD lies between 0 and . PFSs [10] have found the complicated and indeterminacy problems in the assessment detail. Similar to the IFS, the PFS considered the function of MD and NMD , but with the condition that the square sum of MD and NMD must lie between 0 and 1. This means that the PFs are more generalized and comprehensive than the IFS, making them more useful to decision makers. If an expert assigns MD 0.7 and NMD 0.6 to an object, this information does not satisfy the IFS condition, but it is clear that these data can only be conveyed using PFS. As a result, IF decision-making and SMs are distinct from PF decision-making and SMs. Finally, this shows that PFS can covenant more effectively than IFS. Many researchers and investigators have evolved various decision-making methods and techniques [11–15]. Verma [16] defined generalized SMs for PFSs and their applications to MADM. The extended version of the PF-TOPSIS method was studied and applied to MCDM by Zhang and Xu [17]. Peng and Yang [18] introduced the new operations of subtraction and division and studied their fundamental properties. Reformat and Yager built a recommender framework based on collaboration under PFS information in [19]. PFSs are studied by various authors, who introduced various models for group decision-making problems [20–23]. In [24–28], various aggregation operators based on Bonferroni mean (BM) operators, BM with geometric mean, and other aggregation operators were described. Du [29] defined Minkowski-type distance measures for generalized orthopair fuzzy sets. Ejegwa [30] developed a modified Zhang and Xu’s distance measure for PFSs and its application to pattern recognition problems.

Due to the restrictions on the MD and NMD in the IFS and PFS, the information for an object does not clarify, and both concepts have failed to convey the information about an object in certain cases. As a result, in [31], Yager introduced a new concept called *q*-rung orthopair fuzzy set (*q*-ROFS), which expands the MD and MND space by relaxing the condition to sum of the *q*th power of MD and NMD , i.e., , where is equal to or less than 1. This means that if , then *q*-ROFS will be reduced to the IFS, and if , *q*-ROFS will be reduced to the PFS. Clearly, *q*-ROFS is a more generalized framework than the IFS and PFS for explaining unclear and ambiguous details. Liu and Wang described the algebraic sum and product operation for *q*-ROFSs in [32] and studied their application in MCDM. In [33], aggregation operators for *q*-ROFSs based on Maclaurin symmetric mean (MSM) were created. Heronian mean operator [34], partitioned Bonferroni mean operator [35], and power Bonferroni mean operator [36] were developed by the investigator and studied. Liu and Xu developed an MCDM technique for green supplier selection problems under *q*-ROFSs in [37].

The similarity measure (SM) is a powerful method for determining the degree of similarity and dissimilarity between two items in multicriteria decision-making (MCDM) theory and pattern recognition. For the last few decades, the theory and implementations of SMs have been studied. The authors of [38–41] predicted a wide range of SMs for two IFSs. Liu and Cheng’s SMs were modified by Mitchel [42] and also applied to MCDM. Park et al. [43] evaluated the generalized form of fuzzy Hamming distance measure (HDsM), developed IFHDM, and proposed various SMs to apply to the MCDM problem. Torra and Narukawa created some new SMs based on Hausdorff distance and discussed some applications in [44]. Using geometric aggregation operators, Xia and Xu defined the concept of distance measure (DsM) and SM and applied them to the MCDM problem in [45]. Ye [46] developed IF cosine SMs using the cosine feature and applied them to the MCDM problem. Hung [47] specified the likelihood-based SM for the IFS and applied it to medical diagnosis. Shi and Ye provided a modified form of CSMs in [48]. Maoying defined the cotangent SM between two IFSs for MD [49]. Rajarajeswari and Uma described a modified version of cotangent SMs [50], in which they considered MD, NMD, and indeterminacy degrees designated in the IFS. Szmidt introduced the distance measure of IFSs and also developed SMs. In [51–54], some different and generalized distance measures and SMs of IFSs were found. Li and Lu [55] defined some novel similarity and distance measures of PFSs and their applications. Verma and Merigo [56] proposed generalized SMs for PFSs and their applications to MADM. Zeng et al. [57] proposed *q*-rung orthopair fuzzy weighted induced logarithmic distance measures and their application in MADM. Akram et al. [58] defined a novel MCGDM analysis under *m*-polar fuzzy soft expert sets. Akram et al. [59] proposed attribute reduction algorithms for *m*-polar fuzzy relation decision systems. Ali and Sarwar [60] developed a novel technique for group DM under the fuzzy parameterized-rung orthopair fuzzy soft expert framework. Ali and Akram [61] proposed a DM method based on fuzzy *N*-soft expert sets.

Wang et al. [62] defined the SMs for two *q*-ROFSs and applied them to pattern recognition and MCDM problems. Peng and Liu established distance, information, and similarity measures, as well as their relationships, in [63]. The Minkowski-type DM, which includes HM, ED, and Chebyshev DsM, was developed and debated in MCDM by Du [29]. Ali addressed two other methods in [64], developing the notions of possibility and confidence, as well as credibility and certainty, in *q*-ROFSs. The proposed work’s key motivation is to analyze each extension of the fuzzy set (FS) in detail; in intuitionistic fuzzy sets, the two memberships explain the object’s ambiguity, but the IFS fails to explain the described real-life problem. Consider a real-world problem where the MD and NMD values are greater than 0.5, i.e., 0.6 and 0.7; then, , in this case, using the Pythagorean fuzzy sets to describe the real-world problem. When the PFS fails to define some information in a real-world problem, it does not explain the unknown information. As a result, it is failing to clarify uncertainty by using the *q*-rung orthopair fuzzy sets. As a result, the definition of a linear Diophantine fuzzy set (LDFS) was created to cover the importance of membership and nonmembership functions. Raiz and Hashmi introduced the principle of the linear Diophantine fuzzy set (LDFS) in [65], and they demonstrated that it is more generalized than IFS, PFS, and *q*-ROFSs. The IFS, PFS, and *q*-ROFSs are less descriptive and efficient than the LDFS. The key benefit of the LDFS is the availability of comparable parameters (RPs). The MD and NMD have more space than the IFS, PFS, and *q*-ROFSs due to these RPs.

The RPs in the LDFS are constrained and bounded to a finite space, i.e., the number of RPs must be less than or equal to 1. We work on distance and similarity measures of *q*-rung linear Diophantine fuzzy sets to extend the theory and applications of *q*-RLDFS. The distance measure for *q*-RLDFNs will be constructed, and their relations with the similarity measure will be studied. We continue to develop different types of similarity measures and define the Jaccard similarity measure (JMS) of the *q*-RLDFS, which tells us whether the information about the two *q*-RLDFSs is similar or different. For two *q*-RLDFSs, we also expand the JSM to weighted and generalized weighted JSM. Also, for two *q*-RLDFSs, the exponential similarity measure (ESM) was developed and extended for weighted and generalized weighted ESM. The proposed similarity study’s applications are discussed at the end of the article. The proposed similarity measure was used to solve the logistics and supply chain management problem. At the conclusion of the paper, we use the suggested similarity measures to solve a real-world problem of supply chain management. We solve the logistic provider supply chain management problem and show that the proposed SMs are superior to other SMs, currently in use.

The layout is structured as follows: some basic concepts of FS, IFS, *q*-ROFS, and LDFS are offered in Section 2. In Section 3, we define the novel distance measure of *q*-rung linear Diophantine fuzzy sets and the Hamming DM. In Section 4, we introduce a similarity measure of *q*-rung linear Diophantine fuzzy sets and different types of similarity measure such as Jaccard SM and exponential SM. In Section 5, we present the cosine similarity measure for *q*-RLDFSs and cotangent function similarity measure for *q*-RLDFSs. In Section 6, we address the decision frames, a case study for *q*-RLFSs, and a numerical example given to demonstrate the application of the proposed method by using the proposed algorithms. In Section 7, we discuss the comparison between the existing methods and proposed method. In Section 8, this work is eventually outlined.

#### 2. Basic Concepts

The basic concept of fuzzy sets and their extensions are offered in the current section.

*Definition 1. *(see [8]). Let be an arbitrary nonempty set. A fuzzy set (FS) is defined aswhere the function is a mapping from , and for every , , and the function is said to be the MD of in .

*Definition 2. *(see [9]). Let be an arbitrary nonempty set. An IFS in is defined aswhere and are called, respectively, MD and NMD functions such that . Indeterminacy degree can be defined as

*Definition 3. *(see [10]). Let be an arbitrary nonempty set. A *q*-ROFS is denoted by and mathematically defined aswhere and are MD and NMD functions with subject to . The hesitancy MD is denoted by*q*-ROFSs also have certain restrictions on MDs and NMDs.

*Definition 4. *(see [65]). Let be an arbitrary nonempty set. A LDFS is denoted by and mathematically defined aswhere are MD, NMD, and RPs, respectively, and satisfy the condition with . The following is the hesitancy degree:

*Definition 5. *(see [66]). Let be an arbitrary nonempty set. Then, the *q*-rung linear Diophantine fuzzy set (*q*-RLDFS) is represented by and mathematically defined aswhere are MD, NMD, and reference parameters (RPs), respectively. These functions fulfill the restriction , with . The percentage of hesitation can be measured as follows:and *q*-rung linear Diophantine fuzzy number (*q*-RLDFN) is defined as

#### 3. Distance Measure of -Rung Linear Diophantine Fuzzy Sets

In this section, we describe the distance measure (DsM) between two *q*-RLDFSs, as well as their basic properties and their relation.

*Definition 6. *Consider a family of *q*-RLDFSs. Then, a mapping is said to be distance measure (DM), where , if the following conditions hold:(1).(2).(3).(4) is a crisp set.(5)Let . Then, .

Now, we define normalized Hamming distance and normalized Euclidean distance measure between two *q*-RLDFSs and in the following.

*Definition 7. *Consider two nonempty *q*-RLDFNs and on the discrete nonempty set . Then, the normalized Hamming distance measure is denoted by and given as

*Definition 8. *Consider two nonempty *q*-RLDFNs and on the discrete nonempty set . Then, the normalized Euclidean distance is denoted by and given as

We further generalize the HDM and EDM for two *q*-RLDFSs and using a parameter into equations (13) and (14), respectively. The generalized distance measure (GDsM) of two *q*-RLDFSs and is given as

*Remark 1. *If we put , then is reduced to .

*Remark 2. *If we put , then is reduced to .

Next, we use the weight of each with the weight vector subject to such that . Then, we propose a new DsM, so called the generalized weighted distance measure (GWDsM) of and which is denoted by and defined as

#### 4. Similarity Measure of -Rung Linear Diophantine Fuzzy Sets

In this section, we define different types of similarity measure (SM) of *q*-RLDFS and also discuss the basic properties of the proposed SMs. We will use the notion of the Jaccard, exponential, cosine, and cotangent functions and develop the similarity measures between *q*-RLDFSs.

First, we give the definition of similarity measures between two *q*-RLDFSs.

*Definition 9. *Consider a family of *q*-RLDFNs. Then, a mapping is said to be a similarity measure (SM), where , if the following conditions hold:(1).(2).(3).(4) is a crisp set.(5)Let . Then, .

##### 4.1. Jaccard Similarity Measure of -RLDFSs

This section consists of a novel SM between *q*-RLDFSs which is called the Jaccard similarity measure (JSM) between two *q*-RLDFSs. The JSM tells us the information whether the two *q*-RLDFSs are similar or distinct. The JSM provides information of similarity from 0 to 1 for the two *q*-RLDFSs. If the information of similarity of two *q*-RLDFSs is near to 1, then both are similar to each other, and if the information of similarity is near to close to , then both are dissimilar from each other. The JSM is very informative for similarity and dissimilarity for *q*-RLDFSs, so the JSM is useful in the decision-making problem and pattern recognition.(1)(2)(3) iff

*Definition 10. *Consider two nonempty *q*-RLDFNs and on the discrete nonempty set . Then, the Jaccard similarity measure (JSM) is denoted by and given as

Theorem 1. *Consider two nonempty q-RLDFNs and on the discrete nonempty set . Then, defined in equation (11) satisfies the following conditions:*

*Proof. *(1)As are two *q*-RLDFNs and we know that , and Thus, , and summing up to and normalizing, we have . Hence, .(2)The proof of is trivial.(3)Suppose that ; then, . Then, equation (11) becomesConversely, the proof is easy.

Next, we use the weight of each with the weight vector subject to such that . Then, we propose a new SM, called weighted Jaccard similarity measure of and which is denoted by and defined in the following equation:The proposed also satisfies the conditions of Theorem 1.

##### 4.2. Exponential Similarity Measure of -RLDFSs

In this section, the exponential function with Euclidean and Hamming distances between two *q*-RLDFSs and is used to provide a new similarity measure between the two *q*-RLDFSs. Shepard predicted a universal law regarding distance and similarity measures in 1987. Both are related to an exponential function, and using the similarity measure based on the expansion function, the exponential distance measure is defined as

Now, we will talk about the properties of the distance measure, also known as axioms. Every distance measure must satisfy the following axioms:(1)Equality: for every and . Thus, for every and .(2)Minimality: for every . Thus, for every .(3)Symmetry: for every and . Thus, for every and .(4)Triangle inequality: for all and . Thus, we can say that the dissimilarities of any three *q*-RLDFSs satisfy the above properties. We can also say that if is similar to and is similar to , then should be similar to .

Now, in equation (20), we use the normalized Hamming distance (NHD) of two *q*-RLDFSs to get a new kind of SM based on NHD.(1)(2)(3) iff (1)(2)(3) iff

*Definition 11. *Consider two nonempty *q*-RLDFNs and on the discrete nonempty set . Then, ESM based on NHD is generated by using equation (20) and given as

Theorem 2. *Consider two nonempty q-RLDFNs and on the discrete nonempty set . Then, defined in equation (21) satisfies the following conditions:*

*Proof. *(1)We know from the definition of *q*-RLDFS that and . Thus, we can write as and We know about the exponential properties that the zero power of is equal to 1, and the value of the negative power of is less than 1. In this case, we have Thus, we obtained that .(2)The proof of this property is trivial.(3)Suppose that ; then, . Thus, equation (14) becomesHence, .

Next, using the normalized Euclidean distance (NED) of two *q*-RLDFSs in equation (21), we will get the new type of SM based on ED.

*Definition 12. *Consider two nonempty *q*-RLDFNs and on the discrete nonempty set . Then, ESM based on NED is generated. So, the ESM based on NED is given in the following equation:

Theorem 12. *Consider two nonempty q-RLDFNs and on the discrete nonempty set . Then, defined in equation (26) satisfies the following conditions:*

*Proof. *(1)Given that and are the two *q*-RLDFNs and as and , it implies that and We also know about the exponential properties that the zero power of is equal to 1, and the value of the negative power of is less than 1. Then, we have Hence, .(2)The proof of this property is followed by using the symmetric property of NED. So, we omit the proof here.(3)Suppose that ; then, . Thus, equation (26) becomesHence, .

Next, we use the weight of each with the weight vector subject to such that . Then, we propose a new ESM, called the weighted exponential similarity measure (WESM) of and which is denoted by and , respectively.

*Definition 13. *Consider two nonempty *q*-RLDFNs and on the discrete nonempty set . Then, ESM based on NHD and NED is denoted by and , given in equations (31) and (32), respectively.The weighted ESM based on NED is given as

#### 5. Similarity Measure of -RLDFS Based on Cosine and Cotangent Functions

This section is used for the SM based on cosine and cotangent functions for *q*-RLDFSs. The proposed SM is also known as a cosine similarity measure (CSM) and cotangent similarity measure (SM) between two *q*-RLDFSs and . The CSM is generated by using the cosine function, and SM is generated by using the cotangent function. This section has two sections: one is for CSM and another is for SM.

##### 5.1. Cosine Similarity Measure of -RLDFSs

In this section, the authors use the cosine function to develop a CSM for two *q*-RLDFSs and . We also develop the weighted CSM for two *q*-RLDFSs and and also prove some properties of the CSM and WCSM with the distance measure.

*Definition 14. *Consider two nonempty *q*-RLDFNs and on the discrete nonempty set . Then, the cosine similarity measure is denoted by and given asThe CSM of three *q*-RLDFSs , and satisfied the following properties:(1).(2).(3).(4)Let . Then, and .

*Proof. *The first two properties are obvious, so we omit the proof here and prove the last two properties.

(3): assume that ; then, and for all . We have(4): now, to prove the last property, let , so the angle of is larger than the angle of and . Thus, from the fact, we can write as and .

In the following, we present a relation between the distance measure (DsM) and CSM. We explore the DsM between two *q*-RLDFSs using the CSM.

*Definition 15. *Consider two nonempty *q*-RLDFNs and on the discrete nonempty . Then, the DsM between two *q*-RLDFSs is given by the following equation:

Now, we will prove that equation (35) satisfies the conditions of the DsM:(1); then, (2); then, (3); if , then (4)If , then

*Proof. *The first three properties of DsM are obvious; we only prove the last property of equation (35); consider , and such that . Then, DsM of , and is given asTherefore, we can write the following equations from the above equations asNow, consider weight for each with the weight vector for all values of . Then, we propose the weighted CSM for two *q*-RLDFSs.

*Definition 16. *Consider two nonempty *q*-RLDFNs and on the discrete nonempty set . Then, the weighted cosine similarity measure is denoted by and given aswhere is a weight vector of each such that and . If we take , then equation (40) becomes equation (33). The WCSM is a generalized form of the CSM. The WCSM is also satisfied for two *q*-RLDFSs.

The CSM of three *q*-RLDFNs , and satisfied the following properties:(1)(2)(3)

*Proof. *The proof follows from the proof of the properties of equation (35).

Now, we will use the cosine function to propose new SMs and WSMs based on the cosine function. The similarity measure based on the cosine function is given in equations (41) and (42).

*Definition 17. *The SMs based on the cosine function for two *q*-RLDFNs and are denoted by and defined by the following equation:

Also, we prove the following properties for equations (41) and (42):(1)(2)(3) iff (4)If , then and

*Proof. *(1)Since the cosine function values lie between the closed interval 0 and 1 and based on the cosine function, the *q*-RLDF cosine similarity measures are also within