Linear Diophantine Fuzzy Soft Rough Sets for the Selection of Sustainable Material Handling Equipment

: The concept of linear Diophantine fuzzy sets (LDFSs) is a new approach for modeling uncertainties in decision analysis. Due to the addition of reference or control parameters with membership and non-membership grades, LDFS is more ﬂexible and reliable than existing concepts of intuitionistic fuzzy sets (IFSs), Pythagorean fuzzy sets (PFSs), and q-rung orthopair fuzzy sets (q-ROFSs). In this paper, the notions of linear Diophantine fuzzy soft rough sets (LDFSRSs) and soft rough linear Diophantine fuzzy sets (SRLDFSs) are proposed as new hybrid models of soft sets, rough sets, and LDFS. The suggested models of LDFSRSs and SRLDFSs are more ﬂexible to discuss fuzziness and roughness in terms of upper and lower approximation operators. Certain operations on LDFSRSs and SRLDFSs have been established to discuss robust multi-criteria decision making (MCDM) for the selection of sustainable material handling equipment. For these objectives, some algorithms are developed for the ranking of feasible alternatives and deriving an optimal decision. Meanwhile, the ideas of the upper reduct, lower reduct, and core set are deﬁned as key factors in the proposed MCDM technique. An application of MCDM is illustrated by a numerical example, and the ﬁnal ranking in the selection of sustainable material handling equipment is computed by the proposed algorithms. Finally, a comparison analysis is given to justify the feasibility, reliability, and superiority of the proposed models.


Introduction
The multi-criteria decision making (MCDM) techniques have been rigorously investigated by many researchers around the real world. Due to uncertain and vague information, the complexity of human's decision making has grown broadly in the present era. This pursuit gave rise to many resourceful techniques to deal with real-world problems. The methodologies developed for this objective essentially rely on the description of the problem under contemplation. The problem conversely. A q-ROFS is more powerful in growing the freedom between MG and NMG. However, there are some situations when these theories are unable to deal with uncertain information. In order to relax existing constraints on MG and NMG,  introduced the innovative idea of linear Diophantine fuzzy sets (LDFSs). The use of reference or control parameters in LDFS give freedom to DMs in choosing MG and NMG. Moreover, IFSs, PFSs, and q-ROFSs can be considered as specific cases of LDFSs with some limitations (see Figure 1). The semantic comparison of suggested technique with some existing structures is given in Table 1. The goal of this paper is to develop strong models for MCDM that have less limitations than other models. Table 1 shows the advantages and drawbacks of some set theoretical models. The notions of linear Diophantine fuzzy soft rough sets (LDFSRSs) and soft rough linear Diophantine fuzzy sets (SRLDFSs) are established as new hybrid models of soft sets, rough sets, and LDFSs. The suggested models of LDFSRSs and SRLDFSs are more flexible to discuss fuzziness and roughness in terms of upper and lower approximation operators. Certain operations on LDFSRSs and SRLDFSs have been established to discuss a robust multi-criteria decision making (MCDM) for the selection of sustainable material handling equipment. We present four new algorithms based on LDFS, crisp soft approximation spaces, core sets, and reducts.
The organization of this article is provided as follows. Section 2 implies certain fundamental notions of fuzzy sets, IFSs, PFSs, q-ROFSs, and LDFSs. We investigate fascinating operations and score functions of LDFSs. In Section 3, we invent the notions of LDFSRSs and SRLDFSs by applying the LDFS approximation space and crisp soft approximation space. We establish multiple results based on intended structures with the help of illustrations. In Section 4, we present four novel algorithms to determine the material handling equipment selection obstacle. These algorithms are based on the approximation spaces, score functions, upper and lower reducts, and core set. We examine and compare our suggested structures and their results with certain existing notions. Section 5 provides the conclusion of this manuscript.

Set Theories Advantages Semantic Disadvantages
Fuzzy sets [46] Contribute knowledge about Dose not give information about the specific property falsity and roughness of information system Intuitionistic fuzzy sets [47,48] Detect vagueness with Restricted valuation space and agree and disagree criteria does not deal with roughness Pythagorean fuzzy sets [49][50][51] Detect vagueness with larger Cannot handle the roughness of the valuation space than IFSs data and dependency between the grades q-rung orthopair fuzzy sets [52,53] Increase the valuation space of grades For smaller values of q, creates dependency to deal with real-life situations in grades and cannot handle roughness Create independency between the degrees Does not give information about the Linear Diophantine fuzzy sets [54] and increase their valuation space under roughness of information data and cannot the effect of control parameterizations deal with multi-valued parameterizations Rough sets [55] Contain upper and lower approximations of Does not characterize the agree and information dataset to handle roughness disagree degrees with parameterizations Soft sets [56] Produce multi-valued mapping based Does not contain fuzziness and parameterizations under different criteria roughness in optimization Linear Diophantine fuzzy Produce multi-valued mapping based Does not characterize the roughness soft sets (proposed) on the LDF value information system of real-life dataset Contain upper and lower approximations Due to the use of LDFS approximation Linear Diophantine fuzzy soft with LDF degrees under double space for evaluations, it contains rough sets (proposed) parameterizations (soft and reference) and heavy calculations, but easy to handle collect data without any loss of information Use crisp soft approximation space Easy calculations as compared to Soft rough linear Diophantine to evaluate upper and lower approximations LDFSRS, but heavy as compared to fuzzy sets (proposed) with LDF degrees and collect data others (easy to handle) without any loss of information

Some Basic Concepts
First, we assemble fascinating fundamental ideas of LDFSs, rough sets, soft sets, and soft rough sets. Definition 1 ([54]). A linear Diophantine fuzzy set D inQ is defined as: 1] are the satisfaction grade, the dissatisfaction grade, and the corresponding reference parameters, respectively. Moreover, it is required that: The reference parameters are useful for describing objective weights for each pair of MG and NMG. These parameters can be used for multiple objectives to express the physical interpretation of a dynamical system. In addition, γ D (G)π D (G) = 1 − (α D (G)T D (G) + β D (G)S D (G)), whereπ D (G) is called the indeterminacy degree of G to D and γ D (G) is the reference parameter related to the indeterminacy part. It can be seen that the tuples ( T D (G),S D (G) , α D (G), β D (G) ) with G ∈Q are crucial for specifying the LDFS D. Due to this fact, we introduce the new notion of the linear Diophantine fuzzy number (LDFN) denoted asÄ D = ( ṫ D ,ḟ D , α D , β D ) satisfying all the constraints listed above for LDFSs. The collection of all LDFSs inQ is denoted as D (Q).
Example 1 (Combination of drugs in medicine for better treatment.). Medicines are chemicals or compounds used to cure, halt, or prevent disease, ease symptoms, or help in the diagnosis of illnesses. Advances in medicines have enabled doctors to cure many diseases and save lives. A combination drug or a fixed-dose combination (FDC) is a medicine that includes two or more active ingredients combined in a single dosage form. For example: aspirin/paracetamol and caffeine is a combination drug for the treatment of pain, especially tension headaches and migraines. LetQ = {G 1 , G 2 , G 3 , G 4 , G 5 } be the collection of some life-saving drugs. In order to gain a high impact of medicine, two or more drugs can be combined in the preparation of a medicine. If the reference or control parameter is considered as: α = excellent impact against infection produced during surgeries β = no high impact against infection produced during surgeries then its LDFS is given in Table 2. According to the quality, variety, and severity of the disease, a physician provides medicine to the subject. The information data can be classified using control parameters. These parameters represent how much that portion is necessary for the treatment, and their grade values describe how much that factor is present in that medicine. If we change the parameter as: α = "Excellent impact against ear infection" β = "Not highly affective for ear infection" OR α = "Fewer side effects" β = "More side effects", then we can establish various LDFSs that are suitable in other situations. This model helps a pharmacist/doctor/consultant prescribe the most reliable and suitable medicine to the patient for his/her disease. Moreover, reference or control parameters can be used for the purpose of various alternatives in medicine.

Definition 6 ([55]
). Suppose the indiscernibility relation onQ is denoted as R. We assume arbitrarily that R is an equivalence relation. Moreover, Neg R K =Q − K , Pos R K = K , and Bnd R K = K − K are said to be negative, positive, and boundary regions of K ⊆Q. The characteristics of these regions are given as follows: (1) G ∈ Pos R K implies that K certainly contains the elements G ofQ.
(2) G ∈ Neg R K implies that K does not contains the elements G ofQ.
(3) G ∈ Bnd R K implies that K may or may not contain the elements G ofQ.
The equivalence class of object G under the relation R is represented as [G] R . The pair (Q, R) is said to be a "Pawlak approximation space", and R will generate the partitionQ/R = {[G] R : G ∈Q}. Then, pair (R (K), R (K)) is called the rough set of crisp set K, where: are called "lower and upper approximations" of K with respect to (Q, R). If R (K) = R (K), then K is said to be definable; otherwise, it is called a rough set.

Remark 1.
The concepts of the core and reduct in rough set theory are very significant tools in the decision making methods. We can deduce the reduct from the reference setQ. It is used to reduce the unimportant information in the input data. The core is the intersection of all reducts and provides the final optimal decision about the decision making problem (see [3,32]).

Definition 7 ([26]
). For a non-empty collection of alternativesQ and the collection of attributesĠ, the crisp soft relation R ⊆Q ×Ġ is written as: . For a non-empty collection of alternativesQ and the collection of attributesĠ, we have a crisp soft relationÃ ⊆Q ×Ġ. A mappingÃ s :Q → P(Ġ) is written as: The "crisp soft approximation space" is represented by this triplet (Q,Ġ,Ã ). For arbitrary H ⊆Ġ,Ã (H) andÃ (H) are called the "lower and upper approximations", respectively, defined as: The pair (Ã (H),Ã (H)) is called the crisp soft rough set, andÃ ,Ã : P(Ġ) → P(Q) are called "lower and upper approximation operators". P(Ġ) and P(Q) are an assembly of all subsets ofĠ andQ, respectively. IfÃ (H) =Ã (H), thenQ is called definable.

Construction of SRLDFSs and LDFSRSs
In this part, we organize the innovative hybrid structures of soft rough linear Diophantine fuzzy sets (SRLDFSs) and linear Diophantine fuzzy soft rough sets (LDFSRSs) by merging the fundamental compositions of LDFSs, soft sets, and rough sets. In decision making obstacles, we deal with the ambiguities and vagueness in the initial input information. Due to these circumstances, we cannot manage these inputs by utilizing simplistic models. In fuzzy sets, IFSs, PFSs, and q-ROFSs, the opportunities for the assortment of satisfaction and dissatisfaction degrees are restricted due to constraints 0 ≤Ṫ ≤ 1, 0 ≤T +S ≤ 1, 0 ≤T 2 +S 2 ≤ 1, and 0 ≤T q +S q ≤ 1. However, in the LDFS, we can comfortably choose the degrees from [0, 1], due to the reference or control parameters. However, this set does not deal with the vagueness or roughness. We cannot handle uncertainties and parameterizations if we deal only with the roughness of a set. The soft set only works for parameterizations. Therefore, to eliminate these ambiguities and to fill in the research gap, we assemble SRLDFSs and LDFSRSs. These models dispense with the fuzzy degrees, parameterizations, and roughness of the data in the decision making difficulties. The significance of these generalized and authentic notions can be examined in the entire article. Table 3 represents the notations used in the whole manuscript.
) : G ∈Q} where the degrees can be calculated as given in Table 4.

Notation Formulation Notation Formulation
The notions given in Table 4 satisfy the following constraints::

Example 2.
We consider the collection of well known cars given asQ = {G 1 , G 2 , G 3 , G 4 } and the assembly of suitable attributesĠ = {℘ 1 ,℘ 2 ,℘ 3 ,℘ 4 }. The attributes are given as "comfortable and reliable", "good safety", "good maintenance", and "affordable". Let (η,Ġ) be the soft set inQ given as: We consider LDFS, Y D ∈ D (Ġ), given as: The "upper and lower approximations" can be computed by using Definition 9. Upper approximations are given as: Lower approximations are evaluated as: Thus:

Remark 2.
For the "crisp soft approximation space" (Q,Ġ,Ã ), if we take the upper and lower approximations of the following sets listed in Table 5, then we can observe the degeneration of SRLDF approximation operators into different structures based on rough sets.
It is evident from Table 5 that our proposed model is superior and powerful in contrast with other existing structures. However, we cannot decompose the described theories into the SRLDFSs and their respective approximation operators. In simple terms, SRLDFS is the generalization of "soft rough sets, soft rough fuzzy sets, soft rough intuitionistic fuzzy sets, soft rough Pythagorean fuzzy sets, and soft rough q-rung orthopair fuzzy sets".
be "upper and lower approximation operators" over the approximation space (Q,Ġ,Ã ), then the following axioms are true: Proof. See Appendix A. Now, we provide a counter example to prove that equality does not exist in Parts (4) and (8) of Theorem 2.

Approximation Space Set Theories Family of Sets Degeneration of SRLDF After Degeneration of the Approximation Operators Constructed Model
Crisp soft fuzzy subsets ofĠ fuzzy sets [42,43] PF-subsets ofĠ orthopair fuzzy sets Example 3. For the reference setQ = {G 1 , G 2 , G 3 , G 4 } and assembly of decision variablesĠ = {℘ 1 ,℘ 2 ,℘ 3 }, we define a soft set (η,Ġ) inQ written as: The crisp soft relationÃ inQ ×Ġ is given as We can write it as: The "upper approximations" are given as: are "lower and upper approximations" of LDFSs over the "crisp soft approximation space" (Q,Ġ,Ã ) satisfying the following axioms: Proof. The proof is obvious.
Definition 12. For the reference setQ and set of decision variablesĠ, if we define an LDFSRð overQ ×Ġ, are "upper and lower approximations" of Y D about (Q,Ġ,ð) respectively and written as: ) : G ∈Q} where the degrees can be calculated as given in Table 7.

Notation Formulation Notation Formulation
) is a called linear Diophantine fuzzy soft rough set (LDFSRS) in (Q,Ġ,ð). The "lower and upper approximation operators" are represented asð We construct the LDFSR,ð :Q →Ġ, represented in Table 8.  By using Definition 12, we find the "upper and lower approximations" of Y D given by: Similarly, we find all other values for the "upper and lower approximation" of Y D . This implies that:

Remark 3.
For the "linear Diophantine fuzzy soft approximation space (LDFS approximation space)" (Q,Ġ,ð), if we take the upper and lower approximations of the following sets listed in Table 9, then we can observe the degeneration of LDFSR approximation operators into different structures based on rough sets. Table 9. Degeneration of LDFSR approximation operators into different rough set models.

Approximation Space Set Theories Family of Sets Degeneration of LDFSR After Degeneration of the Approximation Operators
Constructed Model fuzzy subsets ofĠ rough sets [42,43] LDF-subsets ofĠ Diophantine fuzzy sets It is evident from Table 9 that our proposed model is superior and powerful in contrast with other existing structures. However, we cannot decompose the described theories into the LDFSRSs and their respective approximation operators. The beauty of this structure is that if we select the "crisp soft approximation space" for LDFSR approximation operators, then it will be degenerated into the proposed SRLDFSs. This generalization provides us a strong relation between both proposed rough set models. In simple terms, LDFSRS is the generalization of "soft fuzzy rough sets, intuitionistic fuzzy soft rough sets, Pythagorean fuzzy soft rough sets, q-rung orthopair fuzzy soft rough sets, and soft rough linear Diophantine fuzzy sets".
Proof. The proof is similar to the proof given in Appendix A.
Proof. The proof is obvious.
Proof. The proof is obvious by following Definition 12.
By using the defined idea of cut sets on LDFSs, we can find the cut sets of LDFSR: where all the calculated cuts are crisp soft relations. Now, we present a result to show that LDFSR approximation operators can be written as crisp soft rough approximation operators.
Theorem 6. Consider that for LDFSR approximation space (Q,Ġ,ð) and D ∈ D (Q), the upper approximation operators can be represented as: 1.
and for arbitrary η, θ ∈ [0, 1], we have: 3. [ Proof. One can conclude the proof of this theorem directly by using Definitions 12 and 14.
Theorem 7. Consider that for LDFSR approximation space (Q,Ġ,ð) and D ∈ D (Q), the upper approximation operators can be represented as: 1.
and for arbitrary η, θ ∈ [0, 1], we have: 3. [ Proof. The proof of this theorem can be obtained directly by using Definitions 12 and 14.

MCDM for Sustainable Material Handling Equipment
The determination of material handling equipment is extremely substantial in the project of an operative industrial system. The efficiency of material flow depends on the selection of appropriate material handling equipment. It promotes capability utilization and increases productivity. Decision support systems and various programs have been developed by various researchers for the selection of the best material handling equipment. In this section, we establish the novel methodologies for the selection of the appropriate and most reliable material handling equipment by using the LDFSRSs and SRLDFSs. The intelligent system, which consists of both technical and economical criteria in the material handling equipment selection process, is presented in Figure 2.

Selection of a Sustainable Material Handling Equipment by Using LDFSRSs
We suppose that a manufacturing company wants to increase efficiency and needs to deal with the materials professionally. The company wants to select that alternative that decreases the lead times and increases productivity. After some basic assessment, the board of the company constructs the set of suitable alternatives given asQ = {G 1 , G 2 , G 3 , G 4 , G 5 , G 6 , G 7 }. To measure the appropriate alternative, several decision makers from the company's technical board are organized. They choose some significant decision variables according to their requirements, given as setĠ = {℘ 1 ,℘ 2 ,℘ 3 ,℘ 4 }, where: ℘ 1 = "Technical: convenience, maintainability, safety required", ℘ 2 = "Monetary: setting up and operational cost, maintenance cost, purchasing cost", ℘ 3 = "Operational: fuel consumption, moving speed, capacity", ℘ 4 = "Strategic: flexibility, level of training required, guarantee".
We divide the attributes into sub-criteria under the effect of parameterizations. This categorizes the data and gives us a wide domain for the selection of truth and falsity grades for the alternatives to the corresponding decision variables. The categorization is given as follows: • "Technical: convenience, maintainability, safety required" means that the alternative is "highly technical" or may be "low". • "Monetary: setting up and operational cost, maintenance cost, purchasing cost" means that the alternative may be "expansive" or "inexpensive". • "Operational: fuel consumption, moving speed, capacity" means that the alternative is "highly operational" or may be "low". • "Strategic: flexibility, level of training required, guarantee" means that the alternative is "highly strategic" or may be "low". Table 10 represents the sub-attributes of the listed criteria. Table 10. Properties of selected attributes.

Attributes Characteristics for LDFSR
"Technical: convenience, maintainability, safety required" ( membership, non-membership , high, low ) "Monetary: operational cost, maintenance cost, purchasing cost" ( membership, non-membership , expansive, cheap ) "Operational: fuel consumption, moving speed, capacity" ( membership, non-membership , high, low ) "Strategic: flexibility, level of training required, guarantee" ( membership, non-membership , high, low ) We developed two novel algorithms (Algorithms 1 and 2) for the selection of best material handling equipment by using LDFSRSs. The flowchart diagram of both algorithms is given in Figure 3.  We use the definitions of score, quadratic score, and expectation score functions for LDFNs A D = ( ṫ D ,ḟ D , α D , β D ) given in [54] and written respectively as: of every alternative inð (B D ) ⊕ð (B D ). 8. Rank the alternatives by using calculated score values. Final decision: 9. Choose the alternative having the maximum score value.

Algorithm 2: Selection of the best material handling equipment by using LDFSRSs.
Input: 1. Input the reference setQ. 2. Input the assembling of attributesĠ.

Construction:
3. According to the necessity of the DM, build an LDFSRð :Q →Ġ. 4. Based on the needs of the decision maker, construct LDF-subset B D ofĠ as an optimal normal decision set. Calculation: 5. Calculate the "LDFSR approximation operators"ð (B D ) andð (B D ) as lower and upper using Definition 12. 6. For "N " number of experts, calculate upper and lower reducts from the calculated "upper and lower approximation operators", respectively. Output: 7. From the calculated "2N " reducts, we get "2N " crisp subsets of the reference setQ. The subsets can be constructed by using the "YES" and "NO" logic. The only alternatives in the reduct having final decision "YES" will become the object of the crisp subset. 8. Calculate the core set by taking the intersection of all crisp subsets obtained from the calculated reducts. Final decision: 9. The alternatives in the core will be our choice for the final decision.

Calculations by Using Algorithm 1
The indiscernibility relation is "the selection of best material handling equipment". This relation can be observed by LDFSR,ð :Q →Ġ given as Table 11.  Thus,ð is an LDFSR onQ ×Ġ. This relation gives us the numeric values in the form of LDFNs of each alternative corresponding to every decision variable. For example, for the alternative G 1 , the decision variable℘ 1 ("Technical: convenience, maintainability, safety required") has numeric value ( 0.73, 0.41 , 0.31, 0.13 ). This value shows that the alternative G 1 is 73% technical and 41% has a falsity value for technicality. The pair 0.31, 0.13 represents the reference parameters for the truth and falsity grades, where we can observe that alternative G 1 is 31% highly technical and it has 13% low technicality. These sub-criteria for the alternatives can be observed from Table 10. All the remaining values can be constructed according to a similar pattern. We consider that experts give some opinion about the attributes and rank them according to their requirement. We convert the verbal description  Table 12.
From Table 12, we can observe that the alternative G 1 is most suitable for the final decision. The bar chart of the ranking results for alternatives is given in Figure 4.

Calculations by Using Algorithm 2
In Algorithm 1, we use the input data in the form of linguistic terms as LDFNs. We only deal with the truth and falsity grades with their reference parameters, and we have no idea about the expert's opinion. Due to the lack of information, we have some uncertainty in our decision. This uncertainty can be removed by giving some weight to the expert's opinion. Therefore, we establish upper and lower reducts for all the experts one by one. The initial five steps of Algorithm 2 are the same as Algorithm 1. We will proceed next by constructing the upper and lower reducts from "upper and lower approximations" of LDFS for all the experts. Suppose that we have three experts from the company's technical committee given as: The reducts from approximations can be constructed by using the following terms. L * = Selection of alternative by using "YES" or "NO", i.e., take average of scores L 3 for all the alternatives. The alternatives that have a greater or equal score L 3 than/to the average can be selected as "YES"; those who have a lesser score than the average value can be neglected as "NO",

F.D = Final decision
The final decision is based on L and L * given in Table 13.            Table 19. The average of the score values of all the alternatives forð (B D ) is 0.572. Table 19. Lower reduct for expert-Z (L Z ) fromð (B D ).
This means that "G 5 " is the most suitable alternative for the final decision.

Selection of the Most Appropriate Material Handling Equipment by Using SRLDFSs
Now, we use our second novel structure of SRLDFS and "crisp soft approximation space" for the selection of the most appropriate material handling equipment. We construct two novel algorithms (Algorithms 3 and 4) for the selection. The flowchart diagram of both algorithms is given in Figure 5.

Algorithm 3:
Selection of the best material handling equipment by using SRLDFSs.

Construction:
3. According to the necessity of the DM, build a crisp soft relationÃ overQ ×Ġ. 4. Based on the needs of the decision maker, construct LDF-subset H ofĠ as an optimal normal decision set.

Output:
7. We use the definitions of the score, quadratic score, and expectation score functions for LDFNsÄ D = ( ṫ D ,ḟ D , α D , β D ) given in [54] and written respectively as: of every alternative inÃ (H) ⊕Ã (H). 8. Rank the alternatives by using calculated score values. 9. Select the object having the highest score value.

Algorithm 4:
Selection of the best material handling equipment by using SRLDFSs.

Construction:
3. According to the necessity of the DM, build a crisp soft relationÃ overQ ×Ġ. 4. Based on the needs of the decision maker, construct LDF-subset H ofĠ as an optimal normal decision set.

Calculation:
5. Calculate the "SRLDF approximation operators"Ã (H) andÃ (H) as "lower and upper approximations" by using Definition 9. 6. For "N " number of experts, calculate upper and lower reducts from the calculated "upper and lower approximation operators", respectively.

Output:
7. From calculated "2N " reducts, we get "2N " crisp subsets of the reference setQ. The subsets can be constructed by using the "YES" and "NO" logic. The only alternatives in the reduct having final decision "YES" will become the object of the crisp subset. 8. Calculate the core set by taking the intersection of all crisp subsets obtained from the calculated reducts.
Final decision: 9. The alternatives in the core will be our choice for the final decision.

Calculations by Using Algorithm 3
We consider the indiscernibility relation "selection of best material handling equipment". This relation is represented as a crisp soft relationÃ overQ ×Ġ given as Table 20.
Thus,Ã overQ ×Ġ is a crisp soft relation. Table 20 shows that we have: We consider that experts give some opinion about the attributes and rank them according to their requirements. We convert the verbal description into the LDFS numeric values in the form of LDFS H. The set H is the LDF-subset ofĠ and written as follows:  Table 21.
From Table 21, we can observe that the alternative G 1 is most suitable for the final decision. The bar chart of the ranking results for alternatives is given in Figure 6.

Calculations by Using Algorithm 4
In this part, we establish upper and lower reducts for all the experts one by one. The initial five steps of Algorithm 4 are the same as Algorithm 3. We will proceed next by constructing the upper and lower reducts from the "upper and lower approximations" of LDFS for all the experts under "crisp soft approximation space". Suppose that we have three experts from the company's technical committee given as: The characteristics and terms for finding the upper and lower reducts are the same as we used in Algorithm 2. Therefore, we directly calculate the reducts for experts.
For expert-Ẋ, the upper reduct of upper approximationÃ (H) (calculated in Algorithm 3) of LDFS H is given as Table 22. The average of the score values of all the alternatives forÃ (H) is 0.629.
This means that "G 1 " is the most suitable alternative for the final decision.

Discussion, Comparison, and Symmetrical Analysis
In this part, we compare our models to the existing approaches and discuss the superiority, authenticity, symmetry, and validity of our proposed structures. The comparison of the proposed structures with existing models is shown in Tables 28 and 29. Such tables reflect the characteristics and limitations of certain current hypotheses. We will observe that our presented models are superior and handle the MCDM techniques efficiently.

Concepts Remarks
Fuzzy set [46] It only deals with the truth values of objects.
Rough set [55] It only deal with the vagueness of input data.
Soft set [56] It only deal with the uncertainties under parameterizations.
Intuitionistic fuzzy set [47] It cannot be applied if 1 <T I (G) +S I (G) ≤ 2 for some G.
Pythagorean fuzzy set [49][50][51] It cannot be applied if 1 <T 2 I (G) +S 2 I (G) ≤ 2 for some G. q-rung orthopair fuzzy set [52,53] It cannot be applied for smaller values of "q" with 1 <T LDFS [54] (1) It can deal with all the cases in which FS, IFS, PFS, and q-ROFS cannot be applied; (2)  We constructed four algorithms based on LDFSRSs, SRLDFSs, and their corresponding approximation spaces. The final results for the decision making problem of material handling equipment selection obtained from these algorithms are given in Table 30. In existing work, the superiority of the proposed model was discussed by examining its degeneration towards some existing rough set models (see Tables 5 and 9). The proposed algorithms are based on the SRLDFSs and LDFSRSs and their approximation operators. Algorithms 1 and 3 are based on the structures with LDFN score values. These algorithms provide us with information about the best and worst alternative. Algorithms 2 and 4 are focused on the core and reducts of the suggested structures. This also involves expert opinion and produces an outcome only for the essential alternative. This does not offer any comparison of the alternatives. Depending on the situation, each algorithm is essential and useful for real-life issues (see Tables 12 and 21).
By using different score functions and evaluating the reducts and core set, we check the behavior of "upper and lower approximations". The final results of Algorithms 1, 3, and 4 are exactly the same. The result of Algorithm 2 is different from the others. This difference is due to the different formulae and different ordering strategies used in the proposed algorithms. As we can see, the three algorithms produce the same decision, so we will go with the alternative G 1 for the final decision. Such structures demonstrate the symmetry in the findings and provide us with an appropriate, ideal approach for the problem of decision making.
Validity test: To demonstrate the validity and symmetry of the results, Wang and Triantaphyllou [36] constructed the following test criteria.
Test Criterion 1: "If we replace non-optimal alternative rating values with the worst alternative then the best alternative should not change, provided the relative weighted criteria remain unchanged". Test Criterion 2: "Process should have transitive nature". Test Criterion 3: "When a given problem is decomposed into smaller ones and the same MCDM method has been applied, then the combined ranking of alternatives should be identical to the ranking of un-decomposed one".
Via these parameters, when we test our results, we see that our findings are correct and reliable and provide us a satisfactory solution to the MCDM problem. Various researchers used numerous techniques based on rough set theory and its hybrid structures to solve decision making difficulties (see [2,3,7,8,[17][18][19]23,24,26,[33][34][35]). Comparing these hypotheses, we found that our proposed models are reliable, efficient, superior, symmetrical, and valid in comparison with those current models.

Conclusions
There are two viewpoints in rough set theory knowledge: positive and axiomatic methods, and it is the same for LDFSRSs and SRLDFSs. This manuscript is a crystal reflection of both aspects of it. We have practiced fundamental ingredients of rough sets, soft sets, and LDFSs and established the proposed structures. With their accompanying illustrations, we provided some findings of such models. Many of the barriers to decision making in the input dataset include unclear, ambiguous, and imprecise details. These models can control these ambiguities better than the fuzzy sets, IFSs, PFSs, q-ROFSs, and LDFSs due to their mathematical formulation, variations, symmetry, and novelty. We introduced several level cut sets of LDFSs and related the recommended approximation operators with these level cut relations. We established various illustrations and results based on LDFSRSs and SRLDFSs approximation operators and corresponding approximations based on level cut sets. We utilized two different approximation spaces to produce variety in the decision making results. We listed the results of the degeneration of the proposed operators and found that our proposed models are generalizations of various existing rough set models. By using approximation spaces, score functions, upper and lower reductions, and core series, we introduced four novel algorithms for the assortment of sustainable material handling equipment. Depending on the situation, each algorithm is essential and useful for solving real-life problems. We discussed the advantages and limitations of the proposed structures with some existing models briefly (see Table 1). In the future, we will expand this research for topological spaces and solve MCDM problems based on the TOPSIS, VIKOR, and AHP families.  Similarly, we can prove the remaining axioms by following these arguments.