Selection of coating material for magnesium alloy using Fuzzy AHP-TOPSIS

Abstract

Magnesium alloys are inherently negative electrochemical potential and are very reactive compared to other engineering metals. They are prone to galvanic corrosion and micro cracks. Various coating materials or Alternatives and the required criteria and sub-criteria for the selection of Alternatives for AZ31B magnesium alloy substrate are identified by means of literature review. Criteria weight and the rank of the alternatives are usually vague and hence uncertainty prevails. The best Alternative from several potential “Candidates”, subject to several criteria and sub-criteria, needs to get decided. In such cases, multi criteria decision making (MCDM) techniques help in determining the MOST suitable coating material. This paper concentrates on the selection of coating material for the magnesium alloy substrate. The problem is subjective, uncertain and equivocal in nature. Hence in this study, fuzzy analytic hierarchy process (AHP) is applied to obtain the weights of criteria and technique for order performance by similarity to ideal solutions (TOPSIS) is utilised for ranking the Alternatives.

1 Introduction

In solving a multi criteria decision making (MCDM) problem, the decision environment affects the decision outcome in which the criteria knowledge is known or uncertain. The decision-making environment can be classified into three types: certainty, uncertainty, and risk [1].

• a. Certainty: In this environment, a decision maker (DM) is fully aware of the criteria which can be quantified by means of numbers.

• b. Uncertainty: Uncertain environment means, the DM has only less knowledge about the criteria at the time of assignment.

• c. Risk: From the historical data, the risk factors can be identified and the necessary steps can be taken.

Zimmerman [2] proposed that fuzzy sets can be used to model uncertainty. AZ31B Magnesium alloy suffers from corrosion attack in spite of its physical deposition treatment on various applications. Hence thermal coating method has been decided to adopt to reduce the intermetallic corrosion. To find the suitable coating material for the alloy, integrated fuzzy analytic hierarchy process–Technique for order performance by similarity to ideal solutions (AHP-TOPSIS) method is being employed.

1.1 Fuzzy logic

Unlike usual “True or False” procedure, “Degrees of Truth” is being adopted by fuzzy logic for finding solutions that are uncertain. Fuzzy logic is like crisp logic in many ways. While crisp sets take the values 0 or 1, Fuzzy sets accept input values that range between 0 and 1. Hence the membership function becomes μ_c:X → [0,1] [3]

1.2 Fuzzy composition

If we represent P as a fuzzy relation from X to Y and Q from Y to Z respectively, the configuration of P and Q is a Fuzzy relation that is described as

µ_PoQ (x_i, z_k) = max (min (µ_P (x_i, y_j), µ_Q (y_j, z and _k))).

The triangular function represented by x(a, b, c) has three parameters ‘a’ (min), ‘b’ (mid) and ‘c’ (max) and trapezoidal function represented by x(a, b, c, d) has four parameters and ‘a’ (min), ‘b’, ‘c’ (essential) and ‘d’ (max) that determine the triangular or trapezoidal shape. Figure 1 represents the triangle and trapezoidal functions.

Figure 1

Example of a typical fuzzy membership and properties.

The triangular and trapezoidal functions are described as shown in Eq. (1.1) and Eq. (1.2)

$$ \begin{aligned} & 0,x \le a \\ & \left( {x - a} \right)/\left( {b - a} \right),x \in \left( {a,b} \right) \\ & \left( {c - x} \right)/\left( {c - b} \right),x \in \left( {b,c} \right) \\ \end{aligned} $$

(1.1)

$$ \begin{aligned} & 0,x \ge c \\ & 0,x \le a \\ & \left( {x - a} \right)/\left( {b - a} \right),x \in \left( {a,b} \right) \\ & 1,x \in (b,c) \\ & \left( {d - x} \right)/\left( {d - c} \right),x \in \left( {c,d} \right) \\ \end{aligned} $$

(1.2)

1.3 Linguistic variables and linguistic values

Linguistic variables are those values that can be conveyed in the way of spoken language. Fuzzy sets always represent imprecise terms. Let L, M and H represent three fuzzy sets that have the member ship functions, µ_Z, µ_M, and µ_H respectively. They are referred as less, medium and high. Fuzzy logic is shown in figure 2 [4].

Figure 2

Working steps of Fuzzy Logic.

1.4 α-Cuts for fuzzy sets

Fuzzy sets can be decomposed into classical sets of weighted combination by applying the principle of identity of resolution. Alpha (α) cuts connects fuzzy sets and crisp sets. α-cut ^αS = {x/S(x) ≥ α} and is inclusive of all the constituents of the universal set X whose membership grades in (S) is either ≥ α.

2 Method

A critical limitation for the extensive application of magnesium alloys is their susceptibility to corrosion. Many processes like effective addition of alloying elements, control of microstructure through rapid solidification, various surface modification treatments, etc have been adopted to control the corrosion. Among these methods, thermal spraying process on the magnesium alloy substrate seems to enhance the corrosion resistance effectively. Hence for thermal spray process suitable coating material is to be identified for the AZ31B magnesium alloy substrate.

Since coating material selection problem belongs to MCDM category, an integrated Fuzzy AHP-TOPSIS is being employed for the solution procedure. TOPSIS can be used as an integrated tool with any other research techniques. It works best with fuzzy AHP as criteria weights are calculated by AHP technique and final ranks of alternatives are obtained by applying TOPSIS. The steps involved are shown in figure 3 [5].

Figure 3

Steps in model development using Fuzzy AHP-TOPSIS integration.

The assumptions of the model development are given in section 2.1. The fuzzy judgment matrix is constructed in section 2.2 and the fuzzy performance matrix is obtained in section 2.3. Execution of defuzzification in section 2.4, is to develop the crisp performance by the concepts of α-cut method and β-risk index. TOPSIS method is applied to obtain the priority ranking order for each coating material alternative in section 2.5 [6].

2.1 Assumptions

This research work considers the scenario for selection of the suitable coating material from enlisted alternatives. The decision makers have to select the best material from several candidate alternatives that work under the same environmental conditions.

In the proposed approach, the evaluation matrix and the weight vectors are defined using the triangular fuzzy numbers (TFN). This is useful in final pair wise comparison of criteria using the sub-criteria evaluation score generated primarily. Table 1 shows the TFN for the judgment matrix.

Table 1 Membership function of the triangular fuzzy number.

Five scales are detailed below. The membership function of the triangular fuzzy number$\tilde{n}$ is defined as

$$ \upmu_{\text{S}} \left( {\text{n}} \right) = \left\{ {\begin{array}{*{20}l} 1 \hfill & {{\text{If}}\;{\text{n}}\;{\text{belongs}}\;{\text{to}}\;{\text{S}}} \hfill \\ 0 \hfill & {{\text{If}}\;{\text{n}}\;{\text{does}}\;{\text{not}}\;{\text{belong}}\;{\text{to}}\;{\text{S}}} \hfill \\ \end{array} } \right. $$

(2.1)

While executing the fuzzy judgment matrix process, these triangular fuzzy numbers $\tilde{1}$, $\tilde{3}$, $\tilde{5}$, $\tilde{7}$, $\tilde{9}$ represent the following linguistic terms as tabulated in table 2.

Table 2 Lexical term and the fuzzy ratio scale.

2.2 Formation of fuzzy judgment matrix

The first step after assumptions that have been made is to determine fuzzy judgment matrix. The steps included are (a) MCDM problem formulation followed by hierarchical structure construction of the problem and (b) Alternative performance determination

2.2a Construction of work break down structure After defining all potential alternatives, required criteria and sub-criteria of the problem, a hierarchical structure has to be constructed. Bottom-Up evaluation criteria have been employed and firstly each potential candidate is measured by means of sub-criteria. Sub-score is assigned to each criterion. The following sections explain the calculation procedures [4].

2.2b Evaluation of tangible sub-criteria The Fuzzy ratio scales for each tangible sub-criterion is created as shown in table 3.

Table 3 Fuzzy ratio scales for a positive tangible sub-criterion.

The following rules are considered:

For a positive sub-criterion, a relatively large fuzzy number will be assigned to the relative high interval value.

If it is a negative sub-criterion, a relatively small fuzzy number will be assigned to the relative high Interim Value.

A fuzzy ratio scale represents a sub score ($\tilde{G}_{{\varvec{ijk}}}$). This means, the Alternative’s (A_i) sub score with respect to each sub-criterion ($c_{jk}$).

2.2c Evaluation of intangible sub-criteria Intangible sub-criteria are difficult to calculate objectively. In order to get a consistent and precise outcome from the decision maker’s subjective judgments, a group decision method has been proposed so that each decision maker (D_s) can grade individual alternative (A_i) on the same sub-criterion ($c_{jk}$). By following this procedure, an alternative can acquire several grades $\tilde{G}$(_ijks) as shown in table 4.

Table 4 Grades (${\tilde{\text{G}}}$_ijks) of Alternative (Ai) as per DM (Ds) on sub-criterion (C_jk).

The above grades are composed in to synthetic sub-score ($\tilde{G}_{{\varvec{i}jks}}$) by Eqs. (2.2)–(2.6)

$$ \tilde{G}_{\text{ijks}} = ({\text{L}}_{\text{ijks}}, \;{\text{M}}_{\text{ijks}}, \;{\text{U}}_{\text{ijks}} ) $$

(2.2)

$$ {\text{L}}_{\text{ijk}} = \hbox{min} \left( {{\text{L}}_{\text{ijks}} } \right),{\text{s}} = 1,2, \ldots , $$

(2.3)

$$ M_{ijk} = \sum\nolimits_{s = 1}^{t} {{\text{M}}ijk} \;{\text{s}} = 1,2, \ldots \ldots ,{\text{t}} $$

(2.4)

$$ {\text{U}}_{\text{ijk}} = \hbox{max} ({\text{U}}_{\text{ijks}} ),{\text{s}} = 1,2, \ldots ,{\text{t}} $$

(2.5)

$$ \tilde{G}(_{{\varvec{i}jk}} ) = \left( {{\text{L}}_{{\text{ijk}}}, {\text{M}}_{{\text{ijk}}}, {\text{U}}_{{\text{ijk}}} } \right) $$

(2.6)

2.2d. Attaining the fuzzy evaluation matrix The sub-scores ($\tilde{G}_{{\varvec{i}jk}}$) of every potential candidate (A_i) related to sub-criteria ($c_{jk}$) can be seen in table 5.

Table 5 Sub-scores (${\text{G}}_{\text{ijk}}$) of Alternative (A_i) with respect to the sub-criteria (${\text{c}}_{\text{jk}}$).

To obtain the scores $\tilde{G}_{{\varvec{ijk}}}$ of each alternative related to each criterion, Eq. (2.7) is used.

$$ \tilde{G}_{ij} = \sum\nolimits_{k - 1}^{q} {\tilde{G}_{ijk} } ,\quad i = 1,2, \ldots ,n\quad j = 1,2, \ldots ,m\quad k = 1,2, \ldots ,q $$

(2.7)

From Eq. (2.7), a decision matrix like Eq. (2.8) can be formed.

(2.8)

Weight vector is to be calculated by means of normalization method. All the criteria (C_j) in Eq. (2.8) get normalized through Eq. (2.9). A fuzzy judgment/evaluation Matrix (A) is obtained in Eq. (2.10) following the normalization

$$ \tilde{a}_{ij} = \frac{{\tilde{G}_{ij} }}{{\sqrt {\mathop \sum \nolimits_{i = 1}^{n} \left( {\tilde{G}_{ij} } \right)^{2} } }},\quad j = 1,2, \ldots ,m $$

(2.9)

(2.10)

where $\tilde{a}_{ij}$ represents the evaluation score of Alternatives (A_i) related to criteria (Cj).

2.3 Obtaining fuzzy performance matrix

The collective accomplishment of each coating material with respect to each criterion is formulated in the form of fuzzy performance matrix. It is attained by the multiplication of the fuzzy judgment matrix with its respective fuzzy weight vector. Hence there arises the need for the determination of fuzzy weight vector.

2.3a Obtaining the fuzzy weight vector In order to represent the relative importance among criteria, weight vector is to be defined. A pair wise comparison is required to obtain the weight vector.

Satty’s scale (table 6) 1–9 was used in table 5 by each decision maker (D_s) to carry out pair wise comparison for all criteria as Eq. (2.11i) and Eq. (2.11ii).

(2.11i)

$$ \begin{aligned} & {\mathbf{b}}_{{{\mathbf{jes}}}} = \, {\mathbf{b}}_{{\varvec{jes}}}^{{ - {\mathbf{1}}}} \quad \varvec{if}\;\varvec{j} \ne \varvec{e} \\ & {\mathbf{b}}_{{{\mathbf{jes}}}} = {\mathbf{1}}\quad \varvec{if}\;\varvec{j} = \varvec{e} \\ \end{aligned} $$

(2.11ii)

where

$$ j = 1,2, \ldots m\quad e = 1,2, \ldots m $$

where score ($b_{jes}$) denotes the measurement of relative importance between each criterion by the decision maker D_s.

Table 6 Saaty’s scale.

Thus a comprehensive pair wise comparison matrix (D) is obtained by combining the grades ($b_{jes}$) of all decision makers. The Eqs. (2.12)–(2.15) represent the combination:

$$ U_{je} = \hbox{max} (U_{jes} ),\quad s = 1,2, \ldots t\quad j = 1,2, \ldots m\quad e = 1,2, \ldots m $$

(2.12)

$$ {\text{L}}_{\text{je}} = \hbox{min} \, ({\text{b}}_{jes} ),\quad {\text{s}} = 1,2, \ldots {\text{t}}\quad {\text{j}} = 1,2, \ldots {\text{m}}\quad {\text{e}} = 1,2, \ldots {\text{m}} $$

(2.13)

$$ M_{je} = \frac{{\sum\nolimits_{s = 1}^{t} {b_{jes} } }}{s},\quad s = 1,2, \ldots, t\quad j = 1,2, \ldots, m\quad e = 1,2, \ldots, m $$

(2.14)

$$ \tilde{b}_{je} = \, \left( {L_{je} , \, M_{je} , \, U_{je} } \right),j = 1,2, \ldots m,\quad e = 1,2, \ldots m $$

(2.15)

where a comprehensive score ($b_{je}$) denotes the comparative importance among criteria which is represented in triangular fuzzy numbers.

$$ \tilde{w}_{j} = \frac{{\sum\nolimits_{e = 1}^{m} {\tilde{b}_{je} } }}{{\sum\nolimits_{j = 1}^{m} {\sum\nolimits_{e = 1}^{m} {\tilde{b}_{je} } } }},\quad j = 1,2, \ldots ,m\quad e = 1,2, \ldots ,m $$

(2.16)

Each criterion has its own importance. The following equation is used to calculate relative weight corresponding to each criterion.

(2.17)

The weights of each criterion are solved sequentially by Eq. (2.17) and thereby one obtains a collective fuzzy weight vector (W) as in Eq. (2.18).

$$ {\text{W}} = {\tilde{\text{W}}}_{1} ,{\tilde{\text{W}}}_{2} \ldots \ldots \ldots ,{\tilde{\text{W}}}_{\text{m}} $$

(2.18)

2.3b Synthesization of fuzzy weight vector The overall evaluation scores of each alternative (A_i) related to each criterion ($C_{j}$) are found out in fuzzy judgment matrix. This has been formulated without considering the relative weight between each criterion. The final fuzzy judgment matrix (H) is obtained by multiplying each criterion weight (${\tilde{\text{W}}}_{\text{j}}$) with the corresponding criterion ($C_{j}$). It is shown in Eq. (2.19).

(2.19)

where $\tilde{h}_{ij}$ denotes the Fuzzy performance score of alternative (A_i) with respect to criterion (C_j) using fuzzy triangular numbers ($L_{ij}$, $U_{ij}$, $M_{ij}$).

2.4 Formulation of crisp performance matrix

Crisp performance matrix is obtained by the execution of defuzzification. This is done by the determination of interval performance cut, α, by considering the risk factors also.

2.4a Calculation of the Interval performance matrix α-cut method is applied to obtain the interval performance matrix ( ^H_α ). Each fuzzy performance score ($\tilde{h}_{ij}$) is agglomerated with α-cut to constitute an interval $\left[ {h_{ijl}^{\alpha } ,h_{ijr}^{\alpha } } \right]$ respectively. The values of $\left[ {h_{ijl}^{\alpha } , h_{ijr}^{\alpha } } \right]$ can be found out by Eqs. (2.20) and (2.21), respectively.

$$ h_{ijl}^{\alpha } = {\text{L}}_{ij} +\upalpha({\text{\rm M}}_{ij} - {\text{L}}_{ij} \text{)} $$

(2.20)

$$ h_{ijr}^{\alpha } = {\text{U}}_{ij} -\upalpha({\text{U}}_{ij} - {\text{M}}_{ij} ) $$

(2.21)

where [$h_{ijl}^{\alpha }$, $h_{ijr}^{\alpha }$] denote the respective left and right points of the Triangle range.

The overall interval performance matrix (H^α) can be obtained from Eq. (2.22), shown below. The α value represents the Degree of Confidence of the Experts.

(2.22)

Larger the α value, stronger the degree of confidence of the decision maker. Continuous increase in α value shows that there will be a narrow progress in the interval between $h_{ijl}^{\alpha }$ and $h_{ijr}^{\alpha }$.

Hence it is clear that the evaluation of the decision is always approximate to the most probable value M_ij of the triangular fuzzy numbers ($L_{ij}$, $U_{ij}$, $M_{ij}$) [7].

2.4b Risk index and defuzzification Decision making process is always accompanied by the risk issues. Hence experts also consider a risk index (β) in dealing with the problem. Defuzzification is executed by compounding the Risk Factor in order to obtain the crisp numbers [8].The overall crisp performance matrix (H ^α_β ) can be obtained from Eq. 2.24 through Eq. 2.23.

$$ h_{ij\beta }^{\alpha } = {\varvec{\upbeta}}h_{ijr}^{\alpha } + (1 - {\varvec{\upbeta}})h_{ijr}^{\alpha } ,\quad 0 \le \alpha \le 1;0 \le \beta \le 1 $$

(2.23)

(2.24)

where $H_{\beta }^{\alpha }$ denotes the crisp performance score in which every alternative (A_i) corresponds to all criteria (C_j) under degree of confidence (α) and risk index (β).

2.5 Ranking the alternatives using TOPSIS [9]

Hwang and Yoon [10] framed the MCDM technique, namely, TOPSIS. This structure has been used to finalise the ranking order of the selected coating materials. This approach was employed as its logic is rational and understandable, involves straight computations, permits the pursuit of best potential candidate or alternative for each identified criterion expressed in an analytical form.

TOPSIS is to define two sets of solutions, viz, the most and the least Ideal solution [11]. The positive ideal solution maximises the criteria that are beneficial and minimises those criteria that seem non-beneficial. The negative ideal solution maximises non-beneficial criteria and minimises the beneficial criteria. We have to find the Optimal Alternative which is closest to the solution that is BEST and farthest from the solution that are LEAST. A “Relative Similarity To The Ideal Solution” has been considered in TOPSIS to select the BEST potential candidate in order to avoid the similarity between the defined solutions. The TOPSIS model is calculated as follows.

• (a) Develop a decision matrix (D) for Alternative

$$ {\text{D}} = \left[ {\begin{array}{*{20}c} {X_{11} } & {X_{12} } & \ldots & {X_{1n} } \\ {X_{21} } & {X_{22} } & \cdots & {X_{2n} } \\ \vdots & \vdots & \cdots & \vdots \\ {X_{i1} } & {X_{i2} } & \cdots & {X_{in} } \\ \vdots & \vdots & \vdots & \vdots \\ {X_{m1} } & {X_{m2} } & \cdots & {X_{mn} } \\ \end{array} } \right] $$

(2.25)

where A_i represents the possible alternatives, i = 1, . . ., m;

X_j denotes the criteria corresponding to the performance of alternatives, j = 1,…, n;

and X_ij is a crisp value which indicates the performance rating of each alternative A_i with respect to each criterion X_j.

• (b) Normalisation of decision matrix

Obtain the normalised decision matrix R (= [r_ij]) calculated as

$$ r_{ij} = \frac{{X_{ij} }}{{\sqrt {\mathop \sum \nolimits_{j = 1}^{n} X_{ij}^{2} } }},\quad j = 1, \ldots ,n;\;i = 1, \ldots ,m $$

(2.26)

where Xij is the performance of alternate i to criterion j.

• (c) Obtaining weighted normalized matrix

This matrix can be obtained by multiplying each column of R with its associated weight wj, that has already been calculated by AHP.

Hence, the weighted normalized decision matrix V becomes

$$ {\text{V}} = \left[ {\begin{array}{*{20}c} {V_{11} } & {V_{12} } & \ldots & {V_{1j} } & \cdots & {V_{1n} } \\ {V_{21} } & {V_{22} } & \cdots & {V_{2j} } & \cdots & {V_{2n} } \\ \vdots & \vdots & \cdots & \vdots & \vdots & \vdots \\ {V_{i1} } & {V_{i2} } & \cdots & {V_{ij} } & \cdots & {V_{in} } \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ {V_{m1} } & {V_{m2} } & \cdots & {V_{mj} } & \cdots & {V_{mn} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {w_{1} r_{11} } & {w_{2} r_{12} } & \ldots & {w_{j} r_{1j} } & \cdots & {w_{n} r_{1n} } \\ {w_{1} r_{21} } & {w_{2} r_{22} } & \cdots & {w_{j} r_{2j} } & \cdots & {w_{n} r_{2n} } \\ \vdots & \vdots & \cdots & \vdots & \vdots & \vdots \\ {w_{1} r_{i1} } & {w_{2} r_{i2} } & \cdots & {w_{j} r_{ij} } & \cdots & {w_{n} r_{in} } \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ {w_{1} r_{m1} } & {w_{2} r_{m2} } & \cdots & {w_{j} r_{mj} } & \cdots & {w_{n} r_{mn} } \\ \end{array} } \right] $$

(2.27)

• (d) Determination of the most and least ideal solutions

The following equation can be used to obtain the positive and negative ideal solutions

$$ \begin{aligned} {\text{A}}^{*} & = \{ (\hbox{max} \;{\text{V}}_{ij} |{\text{j}} \in {\text{J}}),(\hbox{min} \;{\text{V}}_{ij} |{\text{j}} \in {\text{J}}^{\prime } ),{\text{i}} = 1,2, \ldots ,{\text{m}}\} \\ {\text{A}}^{ - } & = \{ (\hbox{min} \;{\text{V}}_{ij} |{\text{j}} \in {\text{J}}),(\hbox{max} \;{\text{V}}_{ij} |{\text{j}} \in {\text{J}}^{\prime } ),{\text{i}} = 1,2, \ldots ,{\text{m}}\} \\ \end{aligned} $$

(2.28)

where

$$ \begin{aligned} {\text{j}} & = \{ {\text{j}} = 1,2, \ldots ,{\text{n}}|{\text{j}}\;{\text{belongs}}\;{\text{to}}\;{\text{Benefit}}\;{\text{Criteria}}\} \\ {\text{j}}^{\prime } & = \{ {\text{j}} = 1,2, \ldots ,{\text{n}}|{\text{j}}\;{\text{belongs}}\;{\text{to}}\;{\text{Non - Benefit}}\;{\text{Criteria}}\} \\ \end{aligned} $$

• (e) Determination of distance between the positive and negative ideal solutions for each defined coating material

  $$ {\text{S}}_{\text{i}}^{*} = \sqrt {\sum\nolimits_{ = 1}^{n} {\left( {V_{ij} - V_{j*} } \right)} 2 } \quad {\text{i}} = 1,2, \ldots {\text{m}} $$

  (2.29)
  $$ {\text{S}}_{\text{i}}^{ - } = \sqrt {\sum\nolimits_{ = 1}^{n} {\left( {V_{ij} - V_{j - } } \right)} 2 } \quad {\text{i}} = 1,2, \ldots {\text{m}} $$

  (2.30)

• (f) Estimation of the relative closeness to the PIS and NIS

$$ {\text{C}}_{\text{i}}^{*} = \frac{{{\text{S}}\,{\text{i}} - }}{{{\text{S}}\,{\text{i*}} + {\text{S}}\,{\text{i}} - }}\quad {\text{i}} = 1,2, \ldots ,{\text{m}}, $$

(2.31)

Where $0 \le {\text{C}}_{\text{i}}^{*} \le 1$ that is, an alternative I is closer to A* as C_i* approaches to 1.

• (g) Prioritize or rank the Alternatives

The potential coating materials are ranked with respect to the relative closeness values obtained.

3 Case study

The proposed methodology is applied to any manufacturing industry where the thermal coating technique on magnesium alloy is being employed. The problem is to select the best coating material among the alternatives identified from literature review and field survey. Minimum porosity, optimal hardness, and optimal structure are the rules to be followed (Kulu 2009) in selection of coating. The process parameters like unmelted particles, roughness, bond strength and inclusion also play a part in the selection. Similarly, the other criteria and sub-criteria that are essential for the best alternative selection are determined. Then the Fuzzy AHP –TOPSIS Integration procedures are adopted in the problem as shown in figure 4 [12].

Figure 4

Work break down structure.

3.1 Problem definition

In view of the studies conducted regarding the properties of AZ31B magnesium alloy which has been coated by means of thermal spray technique, especially high velocity oxy fuel process, the following gaps were identified: [13].

• Micro cracks in the splat intersection with the substrate can occur.

• Poor bonding combination of the applied surface layer to the substrate material.

• Appearance of porosity.

• Distortion of the work piece due to thermal effect.

• Corrosion attack of Mg–Al alloys occurs at α-Mg matrix/ intermetallic interfaces.

• Galvanic corrosion between the substrate and coating is a serious problem.

• Twinning process in microstructure enhances the corrosion. Hence a detailed study of the role of twins is required.

• Structural defects present in the coated surface can accelerate corrosion rate.

  Hence to fill all the aforementioned gaps, a suitable coating material is to be identified for the magnesium alloy.

3.2 Applying methodology or strategy for the case study [14,15,16]

Step 1 Obtaining the fuzzy judgement matrix.

An expert survey was conducted by distributing questionnaire to various industries and based on their collective opinion, criteria and sub-criteria were determined. Thus, 6 criteria and 39 sub-criteria were identified. Criteria are as follows: quantitative (Qut), qualitative (Qul), cost (C), quality (Q), coating structure (CS), and risk factors (R) [17,18,19].

Sub-criteria selected are: density, thermal conductivity, thermal expansion coefficient, hardness, modulus of elasticity, elastic recovery, ultimate or critical load, yield stress, melting temperature, H/E ratio, H³/E² ratio, material cost, manufacturing cost, availability, accessibility, wear resistance, coefficient of friction, radiation sensitivity, hardenability, workability, appearance, oxidation resistance, oxidation rate constant, impact resistance. toxicity, adhesion to substrate, bond strength, durability, brittleness, compatibility of the materials, possibility of surface treatment, framed structure, matrix nature, mixed, aging tendency, porosity, geographic allocation, political stability and foreign policy, exchange rate and economic position.

Figure 5 shows the hierarchical structure with various criteria and sub-criteria required for evaluating the best coating material.

Figure 5

Schematic diagram for the combination of fuzzy AHP and TOPSIS of the proposed model.

The explanation of the criteria and the sub-criteria along with the literature is tabulated in table 7 and 8.

Table 7 The selection criteria.

Table 8 The selection sub-criteria.

Calculation of Fuzzy Judgment Score with respect to each criterion is tabulated in table 9, 10 and 11 respectively.

Table 9 The sub scores of all candidates with respect to all sub-criteria.

Table 10 Rating of each coating material with respect to all criteria.

Table 11 The fuzzy judgment scores of each coating material relating to each criterion.

4 Results

4.1 Computation of weight vector

Fuzzy AHP is used to evaluate the fuzzy weight with the help of pair wise comparison technique. It appears to be difficult to avoid the decision –makers’ substantial judgment or assessment. Hence, AHP is employed to solve this situation by a group decision-making technique which is get converted into the fuzzy form. The computations are tabulated in table 12, 13, 14, 15, 16 and 17 respectively [20].

Table 12 Four pair wise comparison matrix.

Table 13 Four pair wise comparison matrix.

Table 14 Four pair wise comparison matrix.

Table 15 Four pair wise comparison matrix.

Table 16 Comprehensive Pair Wise Comparison Score

Table 17 Criteria weights.

4.2 Determining the fuzzy performance matrix

Fuzzy judgement score of each coating material is combined with the weight vector to develop the fuzzy performance score of the respective candidate related to each criterion. The matrix is tabulated in table 18.

Table 18 Fuzzy performance score of each coating material related to each criterion.

4.3 Decision of interval performance matrix

The degree of confidence (α) of the decision maker and the risk factors are considered. Defuzzification is being carried out. The decision makers have decided to take α value as 0.85. The decision matrix is tabulated in table 19.

Table 19 The collective interval performance rate of α-CUT coating material with respect to each criterion.

4.4 Obtaining the crisp performance matrix (H ^α_β )

Risk index (β) is applicable here for the defuzzification process. The decision makers have unanimously decided to keep β = 0.2. Table 20 shows the tabulated matrix.

Table 20 Comprehensive crisp performance matrix.

4.5 Deciding the favourable and detrimental ideal solutions

Here the TOPSIS technique is being employed for ranking the coating material alternatives. The positive ideal solution (PIS) (h ^α+_jβ ) is being considered as the most favourable crisp performance score and the negative ideal solution (NIS) (h ^α-_jβ ) is being treated as the least favourable crisp performance score. (Eq. (2.28) calculates both PIS and NIS).

4.6 Calculation of the separation weigh up of each Alternative from the ideal solutions calculated

The distance between the positive ideal solution and negative ideal solution can be found out from Eqs (2.29) and (2.30), respectively.

4.7 Solution of the net performance indicator for each Alternative

This involves the calculation of “Closeness of Relation” to the ideal solutions for all the coating material alternatives using Eq. (2.31).

4.8 Prioritization of potential candidates

Ranking of the seven alternatives has been carried out and the BEST alternative suitable for the substrate was identified and recommended for further processes.

Table 21 shows the final ranking of the selected alternatives using the ideal solution method.

Table 21 Separation measurement and ranking of each coating material.

5 Discussion

In this work, MCDM technique has been used to find the best coating material. But studies have shown that the selection of suitable alternative can also be done by using ANOVA by identifying the nonsignificant terms in the coating hardness and Young’s modulus models. Predictive modeling approach in conjunction with global optimization procedure can be used to find the optimum combination of coating parameters. The predictions of the response surface methodology models can be compared with the experimental data [21]. Multiobjective optimization of coating criteria can be obtained by means of multiobjective genetic algorithm solver [22]. 316SS coating performed well in some field tests in petroleum plants [23]. A significant increase in wear resistance of coatings is found. It forms a protective passive layer for the base material [24]. This work can be employed with slight modifications using the mathematical models combined with the proposed model.

6 Conclusion

The attribute weights were obtained by Fuzzy AHP and the coating materials were evaluated with TOPSIS. The Fuzzy AHP–TOPSIS combination was made for robust and consistent results. The technique increases the accuracy of decision-making process and saves time to obtain consistent judgement matrices. Advantages of this technique are: material choice established during early-stage of the product development, avoiding later costs and delays, generate idea through a systematic search of materials, apply a repeatable process for validating the results. From the combination, it has been found that 316 SS exhibits better corrosion resistance than the other selected alternatives. The coating will be having low porosity and oxide contents with good hardness. The mean coefficient of thermal expansion of the as-sprayed 316 SS coating will be less. 316 SS coating provides better mechanical support than bare AZ31B substrate. Above all, 316 SS coating material is highly economical and can be used in aggressive environments. In future other multi-criteria methods can be used to select coating material.