An IVTIFN–TOPSIS Based Computational Approach for Pipe Materials Selection

: This paper proposes a multicriteria decision-making (MCDM) approach, coupling intervalued trapezoidal intuitionistic fuzzy number (IVTIFN) with the technique for order preference by similarity to ideal solution (TOPSIS) to facilitate the selection of pipe materials. Their integration can maximize the advantage in better expressing decision maker’s preference on the proposed evaluation criteria by using a bounded limit instead of an exact value, to rank material alternatives based upon their functional, economic and environmental attributes. To reduce possible information overlapping resulted from the criteria, Mahalanobis distance is incorporated into IVTIFN–TOPSIS to improve the selection results. An illustrative example is provided to verify the proposed approach and demonstrate its practical application, in which four common alternative materials, including carbon steel, galvanized steel, polyvinyl chloride (PVC) and high-density polyethylenes (HDPE), are subject to precise selection to determine their adaptability in waste-water piping. The selection result indicates that the plastic materials are superior to the metal materials. In particular, HDPE is the optimal material alternative for waste-water collection and transport.


Introduction
Pipe materials are prone to chemical corrosion or scaling during waste-water collection and transportation, due to complex and harmful compounds contained. If the corrosion intensifies, it may cause failure of piping system, resulting in leakage and possible environmental damage [1]. Selection of appropriate materials is a premise to help with the careful design of a piping system, ultimately to ensure its operation in a safe and reliable way.
Material selection is crucial for engineering design [2]. It is a complex issue, in which a decision maker may encounter a number of conflicting or competing attributes, e.g., that the economic and functional performance of materials to some extent may not match each other [3]. Basically, the selection of materials is to meet the design requirements [4]. Mercer [5] presented that reliability and longevity was critical to selection of pipe materials, in which internal pressure and external loads were the primary criteria to assess their performances. Anojkumar et al. [6] further incorporated the mechanical properties with corrosion resistance into the selection criteria of pipe materials. Zhang et al. [7] took the compatibility of materials with the working fluid into the selection criteria of heating pipes. In addition to materials' functional performances, economic attributes are also important factors in influencing material selection to decrease the manufacturing cost [8,9]. Kayfeci [10] made a selection from five insulation materials based on their market price. Mendrinos et al. [11] evaluated the performance of in linguistic information, by which its predetermined numerical interval may better specify the fuzzification in a fixed bounded limit [42].
This study provides a MCDM-based computational approach, which couples IVTIFN with the TOPSISto aid engineers in the selection of commercially available materials. Mahalanobis distance is incorporated into IVTIFN-TOPSIS, to discriminate similarities of alternative materials by eliminating the highly correlated decision criteria. An illustrative case example is given to demonstrate its actual application. The study is expected to provide insight into sustainable design of waste-water piping system, ultimately to improve its sustainability.

An Indicator System for Pipe Materials Selection
An indicator system for pipe materials selection is established, as shown in Figure 1. These indicators are classified into three categories, indicating materials' functional, economic and environmental attributes, to reflect the most crucial criteria that should be considered in materials selection for a piping-system design.
The functional attribute reflects the specific performance that certain material must satisfy [21]. In this study, a number of typical criteria are identified to examine functional performances regarding pipe materials, in order to ensure reliability and reduce environmental risk during waste-water collection and transport, including tensile strength (C1), elastic modulus (C2), linear expansion (C3), scaling resistance (C4), and corrosion resistance (C5). C1 reflects materials' capability to withstand a maximum load of tensile stress [43]. C2 reflects the ability of materials to resist elastic deformation, while C3 shows materials' resistibility to deformation caused by temperature change [44,45]. Scaling and corrosion resistance play key roles in pipe materials' functional performance to decrease piping failure [46].
Cost plays an important role in materials selection, by which three criteria are considered, consisting of the materials' marketing price (C6), density (C7) and Hazen-Williams roughness (C8). It is worthy of note that C7 and C8 have indirect impacts on the costs of pipeline construction and operation. The pipe material with a larger density may entail a heavier cost on pipeline installation [47]. In addition, the degree of roughness of pipe material is positively correlated with the maintenance cost in sewage transportation [48].
With environmental considerations being gradually immersed into product design, selection of materials focuses on environmental impact throughout a product's entire lifecycle [49]. In such a context, three indicators are proposed to indicate the environmental performance of pipe material: energy consumption (C9), human health risk (C10) and material recyclability (C11). Energy consumption here focuses on the power consumption related to pipeline construction and maintenance, which may contribute to indirect carbon emissions [49]. Health risk indicates the hazard of toxic components released to the workers in the pipe materials' processing and welding [12]. The material recyclability is used to reflect the potentials of waste prevention, thus to reduce landfill disposal of waste materials [50]. Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 13

Intervalued Trapezoidal Intuitionistic Fuzzy Number
In this study, the data corresponding to performances of alternatives and preferences on the criteria are mainly determined by engineering designers, which are generally presented in a linguistic way. Thus, IVTIFN is employed to convert such raw data into fuzzy numbers.
A set of IVTIFN (shown in Figure 2) is defined as A = A , A = (a , a , a ); (a , a , a ) , (a = a = a ). The membership function (μ (x), μ (x)), indicating that what degree the element (x) subordinates to the set, is expressed as follows [51]: x ∈ a , a 0, Otherwise

Intervalued Trapezoidal Intuitionistic Fuzzy Number
In this study, the data corresponding to performances of alternatives and preferences on the criteria are mainly determined by engineering designers, which are generally presented in a linguistic way. Thus, IVTIFN is employed to convert such raw data into fuzzy numbers.
A set of IVTIFN (shown in Figure 2) is defined as , indicating that what degree the element (x) subordinates to the set, is expressed as follows [51]: Appl. Sci. 2019, 9, 5457 5 of 13  Assume that two arbitrary fuzzy numbers are The arithmetic operation for the two fuzzy numbers used in the study is given as follows: Assume that the number of involved engineering designers for the pipe materials selection is e, and the kth designer's fuzzy rating corresponding to the jth alternative material in the ith criteria is Thus, the fuzzy ratings of different designers are aggregated as r ij , given as follows: The obtained aggregated ratings need to be normalized, and the normalization procedures are given as follows: where B denotes the beneficial criteria, r U j3 = max i r U ij3 ; whilst C represents cost criteria, r L j1 = min i r L ij1 .
Appl. Sci. 2019, 9, 5457 6 of 13 The best and worst value of the jth criterion is denoted as x * The normalized rating is: The normalized matrix of ratings is thus constructed: where m indicates the number of material alternatives, n is the number of evaluation criteria. The engineering designers may also apply linguistic remarks, including Excellent (E), Good (G), Fair (F), Poor (P), Very Poor (VP), to classify alternatives' performance with respect to each criterion. This performances of the alternative materials are by analogy to such remarks, and are converted into corresponding fuzzy numbers, shown in Table 1.  Assume the weightings of the jth criteria assessed by the kth designer is z jk = z L jk1 , z L jk2 , z L jk3 , z U jk1 , z U jk2 , z U jk3 . The obtained fuzzy weightings are aggregated as follows: To facilitate TOPSIS application, the obtained weightings shown in Equation (11) have to be normalized as follows: Thus, the weighting matrix is constructed as follows:

TOPSIS
The above normalized ratings of material alternatives and the related weightings are input into TOPSIS to obtain the distances between alternatives and the best/least alternative. These distances are based on Euclidean distance (IVTIFN-TOPSIS (E)) and Mahalanobis distance (IVTIFN-TOPSIS (M)) respectively, to calculate their closeness coefficients for ranking the material alternatives.
The Euclidean distances between the material alternatives and the best/least alternative are specified as follows: The closeness coefficient is: where C Ei = c L Ei1 , c L Ei2 , c L Ei3 , c U Ei1 , c U Ei2 , c U Ei3 . A defuzzification method, center of gravity (COG), is applied to converting fuzzy numbers C Ei into crisp scores C Ei to rank the alternatives, given as follows [52]: The greater C Ei is, the better the corresponding material is. The Mahalanobis distances between alternatives and the best/least alternative are: where A i A * , and A − indicates the ith material alternative, the best alternative and the least alternative, . Σ is the co-variance matrix of A. If Σ is indicated as a singular matrix, the Moore-Penrose generalized inverse matrix Σ −1 is used to replace Σ [53]. The closeness coefficient is: where C Mi = c L Mi1 , c L Mi2 , c L Mi3 , c U Mi1 , c U Mi2 , c U Mi3 . Similarly, the fuzzy numbers C Mi is converted into crisp scores C Mi to rank the alternatives. The greater C Mi is, the better the corresponding material is.

An Illustrative Case Example
The case example is to conduct materials selection for waste-water piping in a newly constructed municipal sewage treatment plant in Chengdu City, China. According to the design requirements, the nominal diameter of the pipeline is 200 mm, which has to tolerate the pressure of 1.6 MPa. Currently, there is a wide variety of commercial pipe materials on the market, including carbon steels, copper, ductile iron, polyethylene (PE), polyvinyl chloride (PVC), pentatricopeptide repeats (PPR), etc. Given their applications to sewage treatment, this study chooses four common pipe materials, i.e., carbon steel (M1), galvanized steel (M2), PVC (M3) and high-density PE (HDPE) (M4) for further precise selection.
A group comprised of five experienced engineers has been involved in the consultation on pipe materials selection, including two project managers from the treatment plant and three engineers who are engaged in the design of sewage collection and transportation system. Their subjective judgments on the performances of the four alternative materials in respect to each criterion are given in Table 2. Table 3 shows the subjective judgments of the five involved engineers on the importance of the proposed criteria. Table 2. Subjective judgments on performances of the alternative materials.  Table 4 shows that the ranking result from the IVTIFN-TOPSIS (E) model is M3 > M4 > M1 > M2. It is clear that two optional plastic materials, i.e., PVC (M3) and HDPE (M4), are the optimal and near-optimal materials. Figure 3 shows that they have excellent performances in most criteria. For instance, PVC and HDPE show better capacities in corrosive resistance (C4) and scaling resistance (C5), which are the key functional premises to ensure the liability of the piping system. In contrast to M3 and M4, they have their own advantages and disadvantages. PVC shows superiorities in mechanical properties (C1 and C2), marketing price (C6) and roughness (C8). Nevertheless, HDPE is more environmentally-friendly and capable of anti-corrosion. By taking the relative importance of criteria into consideration, the most appropriate material for waste-water piping is PVC (M3). M1 performs better than M2 in most of the evaluation criteria, excepting C4 and C5. instance, PVC and HDPE show better capacities in corrosive resistance (C4) and scaling resistance (C5), which are the key functional premises to ensure the liability of the piping system. In contrast to M3 and M4, they have their own advantages and disadvantages. PVC shows superiorities in mechanical properties (C1 and C2), marketing price (C6) and roughness (C8). Nevertheless, HDPE is more environmentally-friendly and capable of anti-corrosion. By taking the relative importance of criteria into consideration, the most appropriate material for waste-water piping is PVC (M3). M1 performs better than M2 in most of the evaluation criteria, excepting C4 and C5.  A correlation test is conducted to examine the correlation among criteria by using Pearson analysis, through which Table 5 shows that a number of criteria are significantly correlated to each other, highlighted by the functional attributes, i.e., among C1 to C5. The major purpose of the materials selection is to ensure reliable piping of waste water, by which anticorrosion and anticlogging are the prerequisites. The criteria C4 and C5 have directly reflected such functional performances. Although C1, C2, and C3 have certain relationships linked with pipe reliability, their performances on waste-water piping are in a lower priority in contrast to C4 and C5. In such context, they are eliminated from the eleven proposed criteria to decrease information overlapping. Mahalanobis distance is further employed to examine whether the ranking results are varied. Table 6 shows that the materials alternative ranking by using IVTIFN−TOPSIS (M) is M4 > M3 > M2 > M1. Compared with the result by using the IVTIFN−TOPSIS (E), it is clear that HDPE is regarded as the optimal material, whilst carbon steel is the least suitable option.

Discussion
Both of the results from the IVTIFN−TOPSIS (E) and the IVTIFN−TOPSIS (M) model indicate that polymers are superior to the metal materials, which are consistent with that of Petit-Boix et al. [54] indicating that plastic materials are suitable for sewage transport due to their excellent performances on anticorrosion. Zhao et al. [12] combined AHP with gray relational analysis to select similar plastic pipes for heating systems, through which PVC showed great advantages in cost saving, and PE was better in its performance on anticlogging. Li further identified that HDPE was commonly used in landfill leachate transport due to its receptivity to highly concentrated organic matter [55]. Such a result may validate the rationality of our ranking results. Anojkumar et al. [6] further identified that there was no significant difference in the application of TOPSIS and VIKOR to pipe materials selection, but the former is much simpler and time-saving in computation. This may reflect the feasibility of our proposed approach.
From the subjective judgment of the interviewed engineers, they attached great importance to the price, corrosion resistance, and scaling resistance regarding the selection of pipe materials. Such focus indicates that cost and functional properties are still the key drivers in materials selection [56]. However, the engineers pay little attention to the environmental attributes of materials, which reflects that environmental management has not taken as a significant criterion in engineering design [49]. With raw materials being increasingly extracted, engineering design has to consider a transition towards sustainability, not only to follow the economic bottom line, but also to improve the environmental performance [57].

Conclusions
This study employs IVTIFN−TOPSIS to select materials for waste-water piping, in which IVTIFN is used to deal with subjective and linguistic information. To overcome information overlapping and overestimation of evaluation criteria, Mahalanobis distance is introduced to improve the computational approach by elimination of highly correlated criteria.
A case example is to verify the model application, to conduct materials selection for waste-water piping in a municipal treatment plant. Except for general attention given to the materials' functionalities, i.e., anticorrosion and anticlogging, the economic and environmental attributes have been taken into consideration, to increase the sustainability of material selection. Four commonly-available commercial materials, including PVC, HDPE, carbon steel, and galvanized steel are taken as the available alternatives. Both of the results from the IVTIFN−TOPSIS (E) and the IVTIFN−TOPSIS (M) model show that plastic pipes are better than the metal material alternatives. In particular, HDPE is the optimal material, whilst PVC is near optimal for the piping system design.
The study is expected to provide insight into sustainable design of waste-water piping system. However, there are limitations remaining in the proposed method. The materials' performances in respect of each criterion are mainly evaluated by the five invited engineers, whose judgments are fully dependent upon their empirical experiences. This further may result in deliberate preferences in the selection process. Future study will center on the quantification of the materials' performances regarding functional, economic and environmental attributes, thus to reinforce the objectivity of the decision making.
Author Contributions: R.Z. was involved in conceptualizing the whole study and writing the whole paper. Y.H. implemented the calculation and analysis. Y.Y. proposed the model. S.G. collected the data.