Precursor Selection for Sol–Gel Synthesis of Titanium Carbide Nanopowders by a New Cubic Fuzzy Multi-Attribute Group Decision-Making Model

1 Introduction

Advanced ceramics have been broadly used in various formidable industrial devices owing to their invaluable properties [5]. Among these advanced ceramics, titanium carbide (TiC) is one of the most famous ones and is broadly used in the construction of dangerous contraptions, wear-proof coatings, aerospace materials, and reinforcing components in composites because of its high liquefaction point, high hardness, high substance impenetrability to corroding agents, and decent electrical conductivity [8]. Many methods have been founded on the high-temperature pathways that are conventionally used to synthesize TiC. However, these methods have some limitations in their suitability of use. To overcome the limitations of old-fashioned procedures, many different synthesis methods to produce TiC nanopowders have been proposed, including grinding, carbothermal decrease [9], synthetic vapor deposition [2], warm plasma combination [16], self-spreading high-temperature synthesis [15], and sol–gel synthesis [18]. Currently, synthesis of TiC nanopowders by using the sol–gel strategy has gained much attention, as this technique includes systems at the atomic level. Furthermore, the technique results in the improvement of properties, such as high compositional homogeneity at low synthesis temperatures.

Thus, a reasonable decision about the carbon source assumes a huge part in the synthesis of TiC nanopowders through the sol–gel strategy [1, 11]. Routinely, while choosing a material whose elements are known, experts apply trial-and-error techniques or utilize the findings of previous researches. These shortcomings can be managed by using a multi-attribute decision-making (MADM) model. On the other hand, the MADM approach is effective in assessing and selecting problems where the decisions contain a set of alternatives and a set of performance attributes [13]. To date, several MADM approaches have already been planned and established to report the material collection problems. Rao [12] planned a procedure that is formed on the basis of a graph theory and a matrix approach that helps in the collection of an appropriate material from among a large number of existing alternative materials. This procedure measures material determination variables, and material appropriateness evaluation assesses the suitability of materials for an expected material choice issue. Chauhan and Vaish [4] employed the VIKOR and TOPSIS method for order performance according to similarity to ideal solution (TOPSIS) methods to measure the relative ranking of magnetic materials. Shannon’s entropy strategy is utilized as a part of the basic leadership handle on which the relative weight of traits was computed. Milani et al. [10] hybridized the entropy method and the TOPSIS technique. In Chatterjee et al. [3], utilized the VIKOR and ELECTRE strategies to rank materials for which a few supplies are being considered simultaneously, and two illustrations were referred to approve these two strategies for MADM. In Shanian et al. [14], employed a revised Simos method with the ELECTRE III optimization model, with the aim to offer a decision aid structure that accounts for both effects. In Jahan and Edwards [6], reported the VIKOR technique, which depends on interval data to address material determination. In 1965, Zadeh initiated the notion of fuzzy sets [17].

Cubic sets were introduced by Jun et al. [7]. Cubic sets are generalizations of fuzzy sets and intuitionistic fuzzy sets, in which there are two representations: one is used for the degree of membership and other is used for the degree of non-membership. The membership function is held in the form of interval, while the non-membership function is used for the normal fuzzy set.

For the universal increase of the use of the cubic TOPSIS method, the ideal theory and the performance of the proposed cubic fuzzy (CF) multi-attribute group decision-making model (MAGDM) are mainly discussed. In this paper, the cubic TOPSIS method, cubic gray analysis set (CGAS), and CF-MAGDM are defined, with many of their properties investigated. We discuss the relationships among the cubic TOPSIS method, CGAS, and proposed CF-MAGDM. In Section 2, we provide some basic concepts about CF sets (CFSs) and their definitions. In Section 3, we introduce the cubic TOPSIS method to determine the weights of attributes based on the cubic positive-ideal solution (CPIS) and the cubic negative-ideal solution (CNIS). Section 4 proposes a new CGAS. In Section 5, we describe the proposed CF-MAGDM and its numerical applications. In Section 6, we give some new applications of the proposed model for the precursor selection problem. Section 7 gives the results and discussion. The last part, Section 8, offers some closing comments.

2 Preliminaries

In this section, the essential meanings of the CS hypothesis, cubic gray relation analysis set (CGRAS), and cubic TOPSIS technique are presented briefly. In light of these essential ideas, another CF-MAGDM is presented.

Definition 1 ([17]): Let H be a creation of discourse. The idea of fuzzy set was presented by Zadeh and defined as follows: J={h, Γ_J (h)|h∈H}. A fuzzy set in a set H is defined as Γ_J : H→I, which is a membership function; Γ_J (h) denotes the degree of membership of the element h to the set H, where I=[0, 1]. The collection of all fuzzy subsets of H is denoted by I^H . Define the relation of I^H as follows:

(∀Γ, η∈IH)(Γ≤η⇔(∀h∈H)(Γ(h)≤η(h))).

Definition 2 ([17]): An Atanassov intuitionistic fuzzy set on H is a set J={h, Γ(h), η(h): h∈H}, where Γ_J and η_j are the membership and non-membership functions, respectively. Γ_J (h): h→[0, 1], h∈H→Γ_J (h)∈[0, 1], η_J (h): h→[0, 1], h∈H→η_J (h)∈[0, 1] and 0≤Γ_J (h)+η_J (h)≤1 for all h∈H, π_J (h)=1−Γ_J (h)−η_J (h).

Definition 3 ([7]): Let H be a non-empty set. By a cubic set in H, we mean the structure F={h, α(h), β(h): h∈H}, in which α is an IVF set in H and β is a fuzzy set in H. A cubic set F={h, α(h), β(h): h∈H} is simply denoted by F=〈α, β〉. Denote by C^H the collection of all cubic sets in the H. A cubic set F=〈α, β〉, in which α(h)=0 and β(h)=1 (resp. α(h)=1 and β(h)=0) for all h∈H is denoted by 0 (resp. 1). A cubic set D=〈λ, ξ〉 in which λ(h)=0 and ξ(h)=0 (resp. λ(h)=1 and ξ(h)=1) for all h∈H is denoted by 0 (resp. 1).

Definition 4 ([7]): Let H be a non-empty set. A cubic set F=(C, λ) in H is said to be an internal cubic set if C⁻(h)≤λ(h)≤C⁺(h) for all h∈H.

Definition 5 ([7]): Let H be a non-empty set. A cubic set F=(C, λ) in H is said to be an external cubic set if λ(h)∉(C⁻(h), C⁺(h)) for all h∈H.

3 Cubic TOPSIS Method

In this section, we apply the cubic set to the TOPSIS method. We define a new extension of the cubic TOPSIS method by using the cubic set.

Step 1: Suppose that a cubic decision-making problem under multiple attributes has m students and n decision attributes. The framework of the cubic decision matrix can be exhibited as follows:

β={〈[B11−, B11+], η11〉  〈[B12−, B12+], η12S〉⋅⋅⋅〈[Bn1−, Bn1+], η1n〉〈[B21−, B21+], η21〉  〈[B22−,B22+], η22〉⋅⋅⋅[Bn2−, Bn2+], η2n〉⋅⋅⋅⋅⋅⋅〈[Bm1−, Bm1+], ηm1〉  〈[Bm2−, Bm2+], ηm2〉⋅⋅⋅〈[Bmn−, Bmn+], ηmn〉}.

Step 2: Construct a normalized cubic decision matrix R=[β_ij ]. The normalized value r_ij is calculated as below.

A cubic set β=(B_ij , η_ij ) in which B_ij =B⁻, B_ij =B⁺ and η_ij =η:

β=〈[B−∑i=1n(Bij−)2, B+∑i=1n(Bij+)2], η∑i=1n(η)2〉.

Step 3: Make the weighted normalized cubic decision matrix by multiplying the normalized cubic decision matrix by its associated weights. The weight vector W=(w₁, w₂, …, w_n ) is collected from the isolated weights w_j (j=1, 2, 3, …, n) for each attribute C_j satisfying ∑j=1nWj=1. The weighted normalized value is determined by

Bj={sinπ×(1+bij−−ηij)4+sinπ×(1−bij++ηij)4−1}×12−1,

where 0≤B_j ≤1, i=1, 2, 3, …, m and j=1, 2, 3, …, n. The entropy weight of the j^th attribute is defined as

wj=1−βjn−∑j=1nβj.

Step 4: Identify the positive ideal solution (α*) and the negative ideal solution (α⁻). The CPIS (α*) and the CNIS (α⁻) are shown as follows:

αi+={(B1+, η1)(B2+, η2)…(Bn+, ηn)={maxi(Bij+)}{mini(ηi)},

αi−={(B1−, η1)(B2−, η2)…(Bn−, ηn)={mini(B1)}{maxi(ηi)}.

Step 5: Estimate the separation measures using the n-dimensional Euclidean distance. The separation of each candidate from the CPIS, qi∗〈[B−, B+], η〉, is given as qi+〈[B−, B+], η〉=〈13n(|Bij−Bj−|+|Bij−Bj+|+|ηij−ηj|)〉.

The separation of each candidate from the CNIS, qi−〈[B−, B+], η〉, is given as qi−〈[B−, B+], η〉=〈13n(|Bij−Bj−|+|Bij−Bj+|+|ηij−ηj|)〉 (Table 1).

Step 6: Calculate the similarities to the ideal solution. This progression determines the similitudes to an ideal solution by using the following equation:

Zi=qi−〈[B−, B+], η〉qi−〈[B−, B+], η〉+qi+〈[B−, B+], η〉.

3.2 Numerical Applications

Let A={A₁, A₂, A₃} be a set of alternatives and C={C₁, C₂, C₃} be a set of criteria. Let B be a set of decision matrix. The decision matrix evaluates each alternative based on given criteria. Each decision maker gives his opinion in the form of a linguistic variable for rating the performance and the importance of each criteria according to the linguistic variable. Suppose that a cubic decision-making problem under multiple attributes has m students and n decision attributes. The structure of cubic decision metric can be expressed as follows:

	M H	E H	V V B		V V B	F	M		E C	M H	M
D1=	F	M G	B ,	D2=	F	H	V H ,	D3=	M G	M H	L
	V V B	M L	V L		V V B	V V G	M G		M L	M L	V V B

	〈[0.20, 0.60], 0.40〉	〈[0.15, 0.95], 0.40〉	〈[0.10, 0.90], 0.15〉
D1=	〈[0.20, 0.80], 0.30〉	〈[0.30, 0.60], 0.40〉	〈[0.25, 0.60], 0.30〉
	〈[0.10, 0.90], 0.15〉	〈[0.20, 0.50], 0.30〉	〈[0.10, 0.75], 0.15〉
	〈[0.20, 0.60], 0.30〉	〈[0.40, 0.80], 0.60〉	〈[0.25, 0.95], 0.40〉
D2=	〈[0.10, 0.90], 0.15〉	〈[0.20, 0.70], 0.50〉	〈[0.40, 0.80], 0.60〉
	〈[0.20, 0.60], 0.25〉	〈[0.25, 0.95], 0.40〉	〈[0.30, 0.60], 0.35〉
	〈[0.05, 0.95], 0.30〉	〈[0.30, 0.60], 0.40〉	〈[0.20, 0.50], 0.30〉
D3=	〈[0.25, 0.60], 0.40〉	〈[0.30, 0.60], 0.40〉	〈[0.25, 0.60], 0.45〉
	〈[0.25, 0.95], 0.40〉	〈[0.25, 0.95], 0.40〉	〈[0.20, 0.60], 0.30〉

Unknown weight

W1S	W2S	W3S
0.4	0.3	0.3

Step 1: In this step, we aggregate the CF-decision matrix D₁, D₂, D₃ based on the opinions of experts after the weight values for the experts are obtained. The evaluated values provided by different experts can be aggregated based on the CFWG operator as below. The aggregated CF-decision matrix can be defined as follows (Table 2).

Table 2:

The Aggregated CF-Decision Matrix.

	〈[0.1319, 0.6886], 0.3419〉	〈[0.2478, 0.7861], 0.4688〉	〈[0.1621, 0.7668], 0.2777〉
B=	〈[0.1736, 0.7602], 0.2916〉	〈[0.2656, 0.6283], 0.4319〉	〈[0.2878, 0.6541], 0.4494〉
	〈[0.1620, 0.8099], 0.2626〉	〈[0.2286, 0.7348], 0.3618〉	〈[0.1711, 0.6561], 0.2602〉

Step 2: Construct a normalized cubic decision matrix (Table 3).

Table 3:

The Normalized Cubic Decision Matrix.

	〈[0.4856, 0.5268], 0.6569〉	〈[0.5774, 0.5063], 0.6396〉	〈[0.4358, 0.6376], 0.4717〉
B=	〈[0.6391, 0.5816], 0.5603〉	〈[0.6189, 0.4046], 0.5893〉	〈[0.7738, 0.5439], 0.7632〉
	〈[0.5964, 0.6197], 0.5046〉	〈[0.5327, 0.4732], 0.4936〉	〈[0.4600, 0.5455], 0.4419〉

Step 3: Construct the weighted normalized cubic decision matrix by multiplying the normalized cubic decision matrix by its associated weights. The weight vector W=(w₁, w₂, …, w_n ) is composed of the individual weights w_j (j=1, 2, 3, …, n) for each attribute C_j satisfying ∑j=1nWj=1. The weighted normalized value is calculated by using the known weight vector (Table 4):

Attribute	W1S	W2S	W3S
Weight	0.3334	0.3328	0.3337

Table 4:

The Weighted Normalized Value.

	〈[0.1618, 0.1756], 0.2190〉	〈[0.1921, 0.1684], 0.2128〉	〈[0.1454, 0.2127], 0.1574〉
Q=	〈[0.2130, 0.1939], 0.1868〉	〈[0.2059, 0.1346], 0.1961〉	〈[0.2582, 0.1814], 0.2546〉
	〈[0.1988, 0.2066], 0.1682〉	〈[0.1772, 0.1574], 0.1642〉	〈[0.1535, 0.1820], 0.1474〉

Positive ideal solution: The CPIS (α*) is shown as follows:

PIS=

〈[0.2130, 0.2066], 0.1682〉

〈[0.2059, 0.1684], 0.1642〉

〈[0.2582, 0.2127], 0.1474〉

Negative ideal solution: The CNIS (α⁻) is shown as follows:

NIS=

〈[0.1618, 0.1756], 0.2190〉

〈[0.1772, 0.1346], 0.2128〉

〈[0.1454, 0.1814], 0.2546〉

Calculate the separation measures using the n-dimensional Euclidean distance. The separation of each candidate from the CPIS qi+ is given in Table 5.

Table 5:

The Separation of each Candidate from the CPIS and CNIS.

qi+	qi−	Zli	Final ranking
〈0.1081〉	〈0.0514〉	〈0.3222〉	3
〈0.0785〉	〈0.0942〉	〈0.5454〉	2
〈0.0631〉	〈0.0944〉	〈0.5993〉	1

Figure 1:

Final Ranking in Table 5.

4 CGRAS

In this section, we apply the cubic set to gray relation analysis (GRA). We define a new extension of the CGRAS by using the cubic set. We also define the CGRAS.

Let X={[(x0−, x0+), x0], [(x1−, x1+), x1],…, [(xi−, xi+), xi], …, [xm−, xm+), xm]} be sequences (candidates) of two sets, where [x0−, x0+, x0] denote the referential sequences and [xi−, xi+, xi] are comparative sequences. Let [x0j−, xoj+, x0j] and x_ij represent the respective value at point/factor j, j=1, 2, 3, …, n for [x0−, x0+, x0] and [xi−, xi+, xi]. The cubic gray relation coefficients α(x0j−, xij)β(xoj+, xij)γ(x0j, xij)of these candidates at point j can be computed by

where ρ is the identification coefficient at different cases: ρ∈[0, 1], i∈l={1, 2, 3, …, m}, j∈J={1, 2, 3, …, n}. By later attaining the cubic gray relation coefficients altogether, the grade of cubic gray relations α(x0j−, xij)β(xoj+, xij)γ(x0j, xij) between [x0, x0−, x0+] and [xi−, xi+, xi] can be obtained as follows:

5 Proposed CF-MAGDM

This section presents an original idea for CF-MADM founded on combining the notions of CFSs, cubic TOPSIS method, and cubic gray relation analysis (CGRA). Through CFSs, the proposed CF-MAGDM could be effectively dealt with by using vague material data. The steps of the proposed model are as follows.

Step 1: Find the attributes for a certain presentation and shortlist candidates on the basis of the identified attributes satisfying the requirements.

Step 2: CF-decision matrix is obtained as the following form. Particularly, the rating of the candidates, where [B⁻, B⁺] is an IV fuzzy set in X and η is a fuzzy set in X.

Step 3: To terminate the weight of each expert from the cubic decision matrix, a method of entropy weights is offered, as follows:

Step 4: Make an aggregated CF-decision matrix founded on estimations of the authorities after the weight values for the authorities are obtained. The estimated values delivered by different experts can be aggregated, founded on the CFWG operator as below:

q˜ij=(bij, ηij),

The aggregated CF-decision matrix can be defined as follows:

Step 5: Acquire the entropy weights of the attributes. All attributes could not be anticipated to be of equal significance. We characterize a set for the evaluation of significance. To take w, the CF entropy will be used:

where 0≤B_j ≤1, i=1, 2, 3, …, m and j=1, 2, 3, …, n. The entropy weight of the j^th attribute is defined as follows:

Step 6: Make a weighted aggregated CF-decision matrix indomitable based on the different significance of attributes as follows:

Step 7: Determine the CPIS and CNIS as qj+ and qj− by

Let J₁ and J₂ be the benefit attribute and the cost attribute, respectively.

Step 8: Define a CF positive-ideal separation matrix M⁺ and a CF-negative-ideal separation matrix M⁻ as follows:

Step 9: Calculate the cubic gray relational coefficient of each candidate from the CPIS and CNIS as below:

where i=1, 2, …, m and j=1, 2, 3, …, n.

where i=1, 2, …, m and j=1, 2, 3, …, n, and where the identification coefficient ρ=0.5.

Step 10: Compute the degree of gray relational coefficient of each candidate using the resulting equation:

where

Step 11: Estimate the collective index (Zl) based on the degree of gray relational coefficient of each candidate by

Step 12: Decide the primacy of candidates B_i (i=1, 2, …, m) by the proposed Z (i=1, 2, …, m). If any of the candidates has the highest Zl_i values, then it is the most important one. The flowchart of the proposed CF-decision method is illustrated (Figure 2).

Figure 2:

Flowchart of the Proposed Cubic Fuzzy Model.

6 Application of the Proposed Model for the Precursor Selection Problem

TiC powders are commonly synthesized from precursors of Ti and C. The sol–gel method is one of the novel systems for TiC nanopowder synthesis at low refining temperatures. The sol–gel process is a mixed pathway consisting of the preparation of a sol, successive gelation, and solvent removal [15]. In a sol–gel amalgamation pathway, a large volume of titanium alkoxide, for example, titanium superoxide, is condensed in an appropriate dissolving solution. On the other hand, the carbon source is liquefied in water using another measuring glass at room temperature. Thereafter, this course of action is added dropwise to the titanium alkoxide sol. The sol–gel compound should be handled with caution as it contains hazardous substances such as destructive acids and H₂O₂. Shortly thereafter, gelation occurs in which the solvents are vaporized, first at room temperature and thereafter at 110°C, and the dried gels are reinforced in graphite and water using an alumina tube warmer at 1400°C in gushing argon. The warming rate is 10°C/min and the holding time is 1 h. The initial formation temperature of titanium hexachloride (TiC_x O_y ) is under 1000°C, and immaculate TiC powders with a size range of 20–100 nm are finally synthesized. The flowchart of the synthesis of nanocrystalline TiC powders by the sol–gel route is illustrated in Figure 3.

Figure 3:

Schematic Representation of the Preparation of Nanocrystal Line Titanium Carbide Powders by Sol–Gel Route.

Titanium alkoxides are widely used as a titanium source for the synthesis of nanosized TiC by the sol–gel route, while various materials including activated carbon, carbon black, ethyl acetoacetate, sucrose, and sugar can be used as carbon sources. The grain size and morphology of TiC powders can be attributed to the differences in the nature of the carbon obtained from various sources. Therefore, the synthesis of TiC powders by the sol–gel technique largely depends on the appropriate selection of the carbon source as precursor.

In this application example, a research group selects a suitable carbon source as a precursor for blending of TiC nanopowders through the sol–gel amalgamation pathway. There are diverse motivations in the choice of the right carbon source. Five carbon sources are available in the sol–gel process, which are labeled as B₁, B₂, B₃, B₄, and B₅. The clarification of these carbon source candidates is given in Table 6.

In comparative sessions, countless attributes were extracted from a careful assessment of the literature. Then, five attributes for selecting the carbon source were obtained.

• Cost (M₁): The cost of carbon precursors in sol–gel synthesis of TiC nanopowders plays a very significant role in the selection of the best of them. Therefore, one of the main goals in precursor selection in the synthesis of TiC nanopowders is to minimize the cost of the synthesis process.

• Carbon content (M₂): The structure requires little amount of the carbon precursor when it has high carbon content. Furthermore, increasing the carbon content of the carbon source leads to the use of a small amount of water.

• Water solvability (M₃): In the sol–gel process, the carbon foundation is disintegrated in water. The solution is then gently mixed with titanic sol to obtain the gel. The coordination requires less water to accomplish when the carbon source has high water dissolvability, and the volume of included water is standard in the hydrolysis of the sol–gel precursor. Increased water content is related to augmentation in the hydrolysis rate, and the potential to achieve stable sol decreases at higher water/precursor content.

• Temperature of TiC formation (M₄): Decrease in contamination in nanomaterial preparation is one of the fundamental concerns about the innovative work. The fundamental advantage of the sol–gel technique is that nanopowders can be combined at a considerably lower temperature in addition to its lower time and power requirements. Synthesis of TiC at low temperatures is required to obtain TiC nanopowders because at high temperature, the creation rate of valuable nanostones is much quicker and a very high amount of TiC powder is gained.

• Crystallite size (M₅): The important properties of nanomaterials are dependent on continuing the lifespan of precious stones, and observing gem sizes is an important step in product optimization. The expanded correspondence between carbon (from the carbon source) and titanium makes the response reasonable for diminishing the crystallite size of TiC powders. TiC powders with small crystallite sizes in the range of 1–100 nm have been found to show novel and frequently enhanced mechanical properties. The crystallite size from various titanium alkoxide/carbon sources at a strengthening temperature of 1400°C can be measured with X-beam diffraction.

  The categorized arrangement of carbon source selection is illustrated in Figure 4.

Figure 4:

The Hierarchical Structure for the Carbon Source Selection Problem.