Selection of optimal aluminum alloy using TOPSIS method under fuzzy environment

Abstract

Selection of material plays an important role for any industry as an improper selection of material may lead to loss of lives, equipment and money. The present study aims to overcome this kind of problem by finding the best aluminum alloy for industries like aerospace and automotive which are seeks to made parts with this material because of its outstanding properties. A Multi Criteria Decision Making (MCDM) method known as TOPSIS (Technique of Order Preference by Similarity to the Ideal Solution) is utilized under fuzzy environment considering multiple qualitative and quantitative criterion values. A number of decision maker’s opinion is considered with a Triangular fuzzy numbers for weighing the criteria. Best alternative is chosen based on the distance of alternatives from positive and negative ideal solution. It is observed that Al-5083-H112 is the best alternative. Sensitivity analysis is also done to determine how different values of an independent variable will impact a particular dependent variable under a given set of assumptions.

Keywords

Aluminum aerospace and automotive industry fuzzy TOPSIS triangular fuzzy numbers sensitivity analysis

1 Introduction

Aluminum alloy has a wide range of applications in aerospace [1] and automotive industries because of its outstanding property [2] of light weight, corrosion resistance, ease of fabrication and various attracting properties. There are various criteria which influence the material selection in industries where precision is a primary concern. They can be used as stringers, bulkheads, frames, coupling, shafts, gears, fuse parts, nuts, valves, wheel spears etc. Major dominating material that are used for high strength structures are 2xxx and 7xxx series [3]. Improvement is carried out in high strength thick plates, fuselage skin sheets, weight minimization of materials [4]. For proper material selection decision making is utmost important. Multi Criteria Decision Making (MCDM) which is formal approach [5] can used to solve real world problems by making decisions under multiple, conflicting, incommensurate attributes. These are gaining importance as potential tools since 1960’s [6] for analyzing complex real problems. TOPSIS is one of the best MCDM techniques, developed by Hwang and Yoon (1981) [7] which select the best alternative using distance from positive (PIS) and negative-ideal solution (NIS). An alternative which is nearest to the PIS and furthest from the NIS will have the maximum closeness coefficient. Generally most of the problems are formulated based on objective knowledge only, due to the lack of quantitative measures of subjective knowledge and therefore it is impossible to consider them on mathematical based models. Fuzzy logic system introduced by Lotfi A. Zadeh (1965) [8] can be used to incorporate appropriately those subjective or qualitative measures in linguistic terms as fuzzy numbers. A sensitivity analysis can verify whether the selection is robust or not. The idea of sensitivity analysis is to understand the effect on the model with the variation of control variables [9].

A lot of work is carried out in MCDM at different fields considering different situations. A thorough study is carried out [10] on different MCDM techniques to understand the development in different approaches. Two basic MCDM approaches [6] such as Artificial Intelligence (AIMCDM) and Classical (CMCDM) are equally common with wide range of application for selection and evaluation, where fuzzy logic is most popular Artificial intelligence technique used in MCDM. A study on optimal material selection based on entropy method along with TOPSIS [11] proposed that nitride steel is best alternative for exhaust manifold in engineering design where as intelligent MCDM approaches [12] are used for pipes selection in sugar industry. An AHP [13] along with entropy and TOPSIS method is applied for spaceflight mission planning at NASA where Fuzzy MCDM approaches are suggested for future research with more realistic, unquantifiable, imprecise ambiguous and incomplete data. An AHP model [7] weights the criteria and utilizes TOPSIS model [14] under fuzzy environment for ranking of weapon, where the vagueness and ambiguity are represented by triangular fuzzy numbers. A number of decision makers [15] opinions can be considered with fuzzy approach for weight measurement [16] with Quantitative and qualitative attributes [17].

The present study aims to form a model which will select the best alternative material for different parts of aircraft and automotive industries incorporating objective knowledge as well as subjective knowledge. A number of alloys of aluminum are considered as alternatives with different important criteria values. Decision maker’s opinion is c those criterions. One of the best MCDM technique TOPSIS is utilized with fuzzy number system for selection. A Sensitivity analysis is utilized that [18] identify the factors which affect in selection. There are various methods of sensitivity analysis. One of the methods is to interchanging criterions [19] weight which is considered for this problem.

2 Problem description and solving

Present study is conducted based on 8 criteria. Some of the criterion is benefit and some are non benefit criterion. Again these criterions can be considered as subjective and objective. These criterions with their importance are tabulated below in Table 1.

Different alternatives are considered as Al-6061-O (A₁), Al-2024 [20] (A₂), Al-3003-H12 (A₃), Al-5052-H32 (A₄), Al-7075-O (A₅) and AL-5083-H112 (A₆). The important weight of each criterion is calculated from decision maker’s opinion. A TOPSIS MCDM method is utilized under fuzzy environment for selection procedure and sensitivity analysis has been carried out for validation and robustness of the model. A flow chart of the proposed model is provided below in Fig. 1.

2.1 Fuzzy TOPSIS methodology

Fuzzy TOPSIS technique finds widely acceptability due to its sound logic of considering multiple attributes along with quantitative and qualitative values. An algorithm for this problem is shown below [17].

Step 1: Decision makers’ committee with k experts and alternatives (i = 1, 2, …, m) as well as the selection criteria (j = 1, 2, …, n) are chosen properly.

Step 2: Providing important weights for each criterion by different experts according to fuzzy numbers.

Step 3: Calculation of aggregate weights for each criterion (W_j) -

$\begin{matrix} W_{j} & = & (\frac{1}{k}) \times (W_{j 1} + W_{j 2} + \dots + W_{jk}), \\ j = 1, \dots, n \end{matrix}$ (1)

Step 4: Decision matrix is obtained by identifying crisp data as well as linguistic terms.

Step 5: In order to make these criteria unit free and comparable, normalization of is done using the sum of criteria values.

Normalized of benefit related criteria (j = 1, 2, …, n₁) and non-benefit related criteria (j = n₁ + 1, …, n₂) are given in Equation 2. $r_{ij} = {\begin{matrix} \frac{x_{ij}}{\sum_{i = 1}^{m} x_{ij}}, & j = 1, \dots, n_{1} \\ \frac{x_{ij}^{- 1}}{\sum_{i = 1}^{m} x_{ij}^{- 1}} & j = n_{1} + 1, \dots, n_{2} \end{matrix}$ (2)

For fuzzy data having triangular fuzzy numbers (a_ij, b_ij, c_ij), normalization is done as given in Equation 3.

$\begin{matrix} r_{ij} \\ = {\begin{matrix} \begin{matrix} (\frac{a_{ij}}{\sum_{i = 1}^{m} c_{ij}}, \frac{b_{ij}}{\sum_{i = 1}^{m} b_{ij}}, \frac{c_{ij}}{\sum_{i = 1}^{m} a_{ij}}) \\ j = n_{2} + 1, \dots, n_{3} \end{matrix} \\ \begin{matrix} (\frac{c_{ij}^{- 1}}{(\sum_{i = 1}^{m} a_{ij}^{- 1})}, \frac{b_{ij}^{- 1}}{(\sum_{i = 1}^{m} b_{ij}^{- 1})}, \frac{a_{ij}^{- 1}}{(\sum_{i = 1}^{m} c_{ij}^{- 1})}) \\ j = n_{3} + 1, \dots, n_{4} \end{matrix} \end{matrix} \end{matrix}$ (3)

Step 6: Multiplication of aggregate weights of each criterion with normalized criterion values gives weighted normalized decision matrix.

Step 7: Determination of Positive Ideal solution (PIS) $A^{*} = {(a_{1}^{*}, b_{1}^{*}, c_{1}^{*}), \dots, (a_{n}^{*}, b_{n}^{*}, c_{n}^{*})}$ and Negative ideal solution (NIS) $A^{-} = {(a_{1}^{-}, b_{1}^{-}, c_{1}^{-}), \dots, (a_{n}^{-}, b_{n}^{-}, c_{n}^{-})}$ by ranking weighted normalize values for each criterion in descending order.

Step 8: Distance calculation using the distance formula, [7] from the PIS $(D_{j}^{*})$ and from the NIS $(D_{j}^{-})$ for each alternative using Equations 4 and 5. $\begin{matrix} D_{j}^{*} \\ = \sum_{j = 1}^{n} \sqrt{\frac{1}{3} [{(a_{ij} - a_{j}^{*})}^{2} + {(b_{ij} - b_{j}^{*})}^{2} + {(c_{ij} - c_{j}^{*})}^{2}]} \end{matrix}$ (4) $\begin{matrix} D_{j}^{-} \\ = \sum_{j = 1}^{n} \sqrt{\frac{1}{3} [{(a_{ij} - a_{j}^{-})}^{2} + {(b_{ij} - b_{j}^{-})}^{2} + {(c_{ij} - c_{j}^{-})}^{2}]} \end{matrix}$ (5)

Step 9: Calculation of closeness coefficient or proximity of the alternatives (CC_j) to the ideal solution – ${CC}_{j} = \frac{D_{j}}{D_{j}^{*} + D_{j}^{-}}$ (6)

Step 10: Selection and ranking of best alternative with height ${CC}_{j}^{*}$ value.

2.2 Problem solving

A decision committee of 6 members and their linguistic terms are utilized to represent importance of each criterion as very low (VL), low(L), moderate (M), high(H), and very high (VH). Considering their importance of each criterion, triangular fuzzy numbers [7] shown in Fig. 2 is utilized for aggregate weights. Table 2 represents the importance of decision makers.

The reason behind using a Triangular fuzzy numbers is that it is very much effective in formulating decision problems with subjective information and also it is easier to use and calculate.

The membership function for the triangular fuzzy can be defined as Equation 7.

Membership function- $μ_{\tilde{a}} (x) = {\begin{matrix} 0, & x < a_{1} \\ \frac{x - a_{1}}{a_{2} - a_{1}} & a_{1} \leq x \leq a_{2} \\ \frac{x - a_{3}}{a_{2} - a_{3}} & a_{2} \leq x \leq a_{3} \\ 0, & x > a_{3} \end{matrix}$ (7)

Aggregate weights are calculated using Equation 1 and shown in Table 3.

Data of each alternative with respect to each criterion is provided in Table 4. As machinability and corrosion rate does not have any deterministic data linguistic terms are used for representing these two qualitative measures. The triangular fuzzy shown in Fig. 2 is also utilized in assigning fuzzy number for corrosion rate and machinability. Normalization of decision matrix is done using Equations 2 and 3. Weighted normalization is constructed by multiplying aggregate weights with the normalized matrix. Positive Ideal solution (PIS) and Negative Ideal solution (NIS) for each criterion is obtained. The distance of the alternatives from the PIS and NIS are calculated using Equations 4 and 5. Closeness coefficient $({CC}_{j}^{*})$ of the alternatives is calculated using Equation 6 and shown in Table 5.

3 Result and discussion

Distance and closeness coefficients of different alloys according to Fuzzy-TOPSIS MCDM method is tabulated in Table 5.

Arranging the $({CC}_{j}^{*})$ values of alternatives in descending order, the ranking is obtained as shown in Table 6.

It is found that A₆ (Al-5083-H112) got the first rank. Figure 3 shows the ranking of alternatives. Composition of best selected material i.e. Al-5083-H112 has been given in percentage (%) in Table 7.

3.1 Sensitivity analysis

To understand the effect of different criterion weights on selection, a sensitivity analysis is carried out. The concept applied for sensitivity analysis is to interchange two criterions weights in each other while others are constant. This provides total 29 combinations for 8 criteria considering the present condition. This analysis shows the behavior of change in ranking due to weight manipulation. For each combination ${CC}_{j}^{*}$ and ranking has been calculated using present methodology whose results are given in the form of graphs in Figs. 4 and 5.

Figure 4 represents a change in closeness coefficient values with weight manipulation. Alternative 4 and 6 are very close to each other with high values of ${CC}_{j}^{*}$ , where as other alternatives has little low ${CC}_{j}^{*}$ value. Change in rank can be illustrated by Fig. 5, which shows most alternatives has some change in ranking while weight changes. The maximum amount of change in rank is observed in alternative 3, 4 and 6 where alternative 4 and 6 interchanges only in 1st and 2nd rank at different conditions. The least amount of change is observed in alternative 2 while last 3 criterions weight interchanges in eachother.

4 Conclusion

Selection of material is getting more complex as selection need to be based on influencing criterions which are may be quantitative as well as qualitative factors. Mathematical procedures for solving in real time also need to be improved accordingly considering expert opinions. As in this problem incorporation of fuzzy, simplifies the expert opinion as well as non deterministic terms. Decision making using TOPSIS methodology shows that alternative 6 got the 1st rank. This is due the high closeness coefficient of alternative 6. This is because, the distance of PIS is maximum and distance of NIS is minimum for A6 while compared with other alternatives. Again as the weight is assigned by the decision makers this also plays an important role for ranking. Sensitivity analysis represents that excluding the initial solution almost 82.14% of the conditions alternative 6 got the highest ranking where as alternative 4 got the highest rank for 21.43% of the cases. Which validate the solution and verified that A₆ (Al-5083-H112) is best suitable material for making various parts of aircraft and automotive industries. This selection was an overall evaluation of a problem. Further this can be made more specific such as making any particular component while considering various quantifiable and un-quantifiable criterions which have influence in products material selection. Other MCDM techniques such as MOORA, PROMETHEE can also be utilized for validation.

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