**An Extension of the CODAS Approach Using Interval-Valued Intuitionistic Fuzzy Set for Sustainable Material Selection in Construction Projects with Incomplete Weight Information**

#### **Jagannath Roy 1, Sujit Das 2, Samarjit Kar <sup>1</sup> and Dragan Pamuˇcar 3,\***


Received: 21 February 2019; Accepted: 15 March 2019; Published: 18 March 2019

**Abstract:** Optimal selection of sustainable materials in construction projects can benefit several stakeholders in their respective industries with the triple bottom line (TBL) framework in a broader perspective of greater business value. Multiple criteria of social, environmental, and economic aspects should be essentially accounted for the optimal selection of materials involving the significant group of experts to avoid project failures. This paper proposes an evaluation framework for solving multi criteria decision making (MCDM) problems with incomplete weight information by extending the combinative distance assessment (CODAS) method with interval-valued intuitionistic fuzzy numbers. To compute the unknown weights of the evaluation criteria, this paper presents an optimization model based on the interval-valued intuitionistic fuzzy distance measure. In this study, we emphasize the importance of individual decision makers. To illustrate the proposed approach, an example of material selection in automotive parts industry is presented followed by a real case study of brick selection in sustainable building construction projects. The comparative study indicates the advantages of the proposed approach in comparison with the some relevant approaches. A sensitivity analysis of the proposed IVIF-CODAS method has been performed by changing the criteria weights, where the results show a high degree of stability.

**Keywords:** material selection; multiple criteria decision making; IVIFN; CODAS; sustainability

#### **1. Introduction**

Selecting the right material to develop a particular product is essential for each of the organizations to survive in the competitive business sectors. One suitable material can substantially minimize the production cost and maximizes the profit. It is also needed to improve the product performance and customer satisfaction. Due to the presence of varieties types of almost similar kind of materials, the task of material selection has become one of the most challenging tasks in the real life environment [1–3]. Thus, the material selection process is essential in many practical industrial problems like automotive parts manufacturing, selection of robots and forklifts, designing the femoral component of cemented total hip/knee replacement, building constructions, etc. Hence, in the last few decades, several researchers have focused on this domain. This paper deals with the material selection problem with sustainability perspectives for a construction company.

The material selection problem (MSP) in the presence of imprecise and incomplete information is regarded as an important area of research where researchers try to solve ill-structured complicated real

circumstances. Researchers have applied classical fuzzy set theory (CFST) to explore and solve the MSP under uncertain environment where they mainly considered the individual performance of every material. Due to the complicated and ill-structured characteristics and several practical contexts of MSP, it is necessary to adopt/use newer advanced techniques which can flexibly handle uncertain data while assessing the performance of alternative materials. In the recent years, many researchers have proposed novel methods for solving MSP under uncertainty, such as, two interval type 2 fuzzy TOPSIS method [4], interval 2-tuple linguistic model [5], and soft computing tool based on fuzzy grey relations [6]. However, only a few researchers [7] have used interval-valued intuitionistic fuzzy sets (IVIFSs) in MSP under incomplete weight information. The IVIFS theory [8] is more practical and reliable approach than CFST for handling imprecision and fuzziness in DMs' judgments in real world decision making problems. IVIFS considers both the degree of membership and nonmembership value of a component in a given set and they can take interval values rather than exact numbers. When DMs often find it difficult to evaluate the uncertain material performance with just a single valued number using CFST, IVIFS offers more flexibility and DMs get an extra degree of freedom to evaluate the same. Hence, it has become considerably important to explore MSP under uncertainty with more efficient and appropriate mathematical approaches that use the IVIFSs for better handling of imprecise information. Due to its inherent flexible nature, recently many authors have contributed on IVIFS and applied it in decision problems. Some significant contributions of IVIFS in decision making problems are narrated below. Cheng [9] proposed a decision making method to select hotel locations using IVIFS, where the authors used IVIFS to denote the values and weights of the attributes. Wen et al. [10] introduced a new method to aggregate the IVIFNs when they are distributed all over a region. For that purpose the authors studied the interval-valued intuitionistic fuzzy definite integral and based on that they proposed interval-valued intuitionistic fuzzy definite integral (IVIFDI) operator. Rashid et al. [11] studied interval-valued knowledge measure for the IVIFSs by extending the knowledge measure of IFSs and proposed interval-valued information entropy measure for IVIFSs. Wang and Chen [12] proposed a decision making method using IVIFSs, linear programming (LP), and the extended TOPSIS method, where they used LP methodology to obtain optimal weights of attributes. Gupta et al. [13] combined extended TOPSIS and LP method to propose a multi-attribute group decision making (MAGDM) method in the context if IVIFS. Interval-valued intuitionistic fuzzy matrix (IVIFM) was used by Das et al. [14] to propose a decision making approach, where the authors assigned confident weights to the experts and then presented a decision making algorithm [15].

MSP has been recognized as an MCDM problem [7] and one has to take care of several criteria (sometimes referred as objectives) systematically, such as enactment, cost, use, mobility, transportation, availability, disposal, environmental norms, maintenance, etc., while seeking the best material [1–3]. The applicability of MSP in the context of MCDM to design a product/service has been explored in the literature survey segment of this paper. The availability of numerous MCDM tools has helped the DMs to identify and select the optimal choice for their MSP (refer to Section 2.2). The CODAS method is a new evaluation tool, recently proposed by Ghorabaee [16], has been proved to be efficient to deal with MCDM problems. Due to its inherent characteristics, the CODAS method possesses a systematic and simple computation procedure which is logically sound to represent the underlying principle of real life decision making problems. Thus, it has become significant to incorporate CODAS method in MSP for evaluating and ranking the alternative materials under incomplete criteria weight information and IVIFNs context.

The above-mentioned deliberations motivate us to extend the CODAS method for material selection problems (MSPs) under uncertainty. Then the extended CODAS method is used to develop a novel MCDM framework using IVIFNs under difficult situations where criteria weight are partial known or totally unknown. The projected MSP solution procedure is able to replicate both subjective judgments and objective information in practical engineering, manufacturing, and industrial problems in the context of IVIF. Finally, one illustrative example is given to examine the proposed IVIF-CODAS method for MSP followed by a real case study which is demonstrated for a sustainable building construction company. The outcomes of the proposed research framework can help DMs (engineers, manufacturers, and designers) to have an effective decision for complicated MSPs in uncertainty.

The rest of the paper is structured as follows. Section 2 offers a literature review of MSP and sustainability perspectives along with IVIFSs and MCDM. Section 3 discusses the preliminaries on IVIFSs. Next, the extended CODAS method with IVIFNs and incomplete weight information is presented in Section 4. Two illustrative examples are discussed in Section 5 to show the usefulness of the projected evaluation framework. Section 6 explains the outcomes through result comparison and sensitivity analysis while Section 7 concludes the current research work.

#### **2. Literature survey**

#### *2.1. Material Selection in Construction Industry and Sustainability*

The construction industry has been revealed as the fastest growing industry throughout the world due to the constant increase in urban population. This fastest growth has influenced the society for better economic and social movement and simultaneously has triggered the environmental pollution factors. Some researchers [17–19] investigated that the energy used ratio in recent infrastructures is six times more than that of the older ones, especially in United Arab Emirates (UAE). Furthermore, it was documented (https://ccap.org/assets/Success-Stories-in-Building-Energy-Efficiency\_CCAP.pdf) that the construction industry consumes 40% of the world's energy. These factors had inspired many countries to consider eco-friendly infrastructures such as buildings which gives more importance to sustainable construction rather than economic concerns [3]. Since sustainable construction has a direct impact on environment, economy, and society, the construction agencies are continuously trying to adopt it in their work cultures mainly in the form of sustainable design, structure, and material selection. This kind of changed scenario has drawn the focus of many researchers and consequently, several research works have been carried out on sustainable construction. As a pioneer of this concept, Kibert [20] stated that "Sustainable construction is the creation and responsible management of a healthy built environment based on resource efficient and ecological principles". Among many other sustainable construction factors, sustainable material selection imparts a key role and directly effects building sustainability. Radhi [21] performed an experiment on UAE construction and found the impact of UAE construction on global warming. Elchalakani and Elgaali [22] combined the effects of recycled aggregate and recycled water and discussed the strength and durability of recycled concrete. The authors prepared a moderate strength concrete using recycled water and recycled aggregate obtained from construction wastes. Al-Hajj and Hamani [23] investigated the existing studies regarding the sources of waste and suggested some measures to reduce it. The authors noticed that the lack of awareness and poor design were the main causes of material waste. Despite the need to explore MSP in construction projects under sustainability perspectives, only a few research papers [18,24,25] are found in the literature. A more extensive literature survey can be found in [18,23]. As per our knowledge, no researcher has explored the MSP with sustainability norms in the Indian scenario. Hence, this paper attempts to fill this gap and enrich the literature of MSP in sustainable construction projects.

#### *2.2. MSP and Various MCDMs*

To find the optimal choice of MSP in construction and design engineering, one can consider diverse methodologies, such as MCDM techniques, statistical approaches, artificial intelligence, mathematical programming, and hybrid methods. Among them, MCDM tools are most widely used for MSP, since they can easily and successfully solve the evaluation problems that are complex and have multiple conflicting objectives/criteria. For example, Bahraminasab and Jahan [26] applied a comprehensive VIKOR (VIsekriterijumska optimizacija i KOmpromisno Resenje) method to find the best material for a femoral component of total knee replacement. A new framework was developed by Jahan et al. [27] for weighting of criteria in MSP. Chauhan and Vaish [28] proposed a hybrid evaluation model including entropy, VIKOR, and TOPSIS (Technique for order preference by similarity to ideal solution) methods to select the soft and hard magnetic materials. Interval 2-tuple linguistic VIKOR [3], multi-objective optimization on the basis of ratio analysis [29] (MOORA), complex proportional assessment [30] (COPRAS) are also successfully used in MSP. In recent years, Anojkumar et al. [31] used fuzzy AHP (Analytic Hierarchy Process), VIKOR and TOPSIS for MSP in the sugar industry. Furthermore, interval type 2 fuzzy TOPSIS [4], Multi-Attributive Border Approximation area Comparison [7] (MABAC), hybrid MCDM framework with DEMATEL, ANP and TOPSIS models [18], neutrosophic MULTIMOORA [32], and grey-correlation-based hybrid MCDM method [33] are notable contributions in MSP in recent times. All the authors except Xue et al. [7] have used either fuzzy sets or crisp sets.

Although many material selection methods are available in the literature, still there is a need to explore the MCDM techniques by incorporating IVIFS theory. IVIFNs remove the limitations of CFST and offer a more rational and computational flexibility to address uncertainty and ambiguity in data. IVIFS theory is receiving much interest from researchers and has been effectively used in a diverse domain of real life problems. This motives us to extend the CODAS method with IVIFNs and apply it to two real problems with incomplete criteria weight information. In this article, the readers will find a systematic and comprehensive research framework for MSP, which is capable of discoursing with the subsequent research questions: (1) what are the sustainability indicators for MSPs in construction projects in India? (2) How to set priorities of these indicators in the evaluation process? (3) Which should be the optimal choice for a sustainable alternative (here, brick) for building construction?

#### **3. Preliminaries**

This section reviews the related ideas. Interval-valued intuitionistic fuzzy set (IVIFS) [8] is a generalization of intuitionistic fuzzy set (IFS). Compared to IFS, where the values of membership and non-membership functions are exact numbers, an IVIFS has the characteristic that those values are in intervals [0, 1] instead of exact numbers. Hence, IVIFS is more suitable in uncertain situations.

**Definition 1.** *Let X be a universal set. An IVIFS A in X is expressed as*

$$\mathbf{A} = \left\{ \left< \mathbf{x}, \left[ \mu^{\mathrm{l}}\_{\mathrm{A}}(\mathbf{x}), \ \mu^{\mathrm{r}}\_{\mathrm{A}}(\mathbf{x}) \right], \ \left[ \mathbf{v}^{\mathrm{l}}\_{\mathrm{A}}(\mathbf{x}), \ \mathbf{v}^{\mathrm{r}}\_{\mathrm{A}}(\mathbf{x}) \right] \right> \, \Big|\, \mathbf{x} \in \mathsf{X} \right\} \tag{1}$$

*where μl <sup>A</sup>*(*x*), *<sup>μ</sup><sup>r</sup> <sup>A</sup>*(*x*) <sup>∈</sup> [0, 1] *and νl <sup>A</sup>*(*x*), *<sup>ν</sup><sup>r</sup> <sup>A</sup>*(*x*) ∈ [0, 1] *are respectively the interval-valued degrees of membership and non-membership of an element x* ∈ *X to A, and the sum of upper bounds of these two interval-valued degrees is not greater than 1,* <sup>0</sup> ≤ *<sup>μ</sup><sup>r</sup> <sup>A</sup>*(*x*) + *<sup>ν</sup><sup>r</sup> <sup>A</sup>*(*x*) ≤ <sup>1</sup>*. If <sup>μ</sup><sup>l</sup> <sup>A</sup>*(*x*) = *<sup>μ</sup><sup>r</sup> <sup>A</sup>*(*x*) *and νl <sup>A</sup>*(*x*) = *<sup>ν</sup><sup>r</sup> <sup>A</sup>*(*x*), ∀*x* ∈ *X*, *then the IVIFS A* = *x*, *μl <sup>A</sup>*(*x*), *<sup>μ</sup><sup>r</sup> <sup>A</sup>*(*x*) , *νl <sup>A</sup>*(*x*), *<sup>ν</sup><sup>r</sup> <sup>A</sup>*(*x*) <sup>∨</sup> *<sup>x</sup>* <sup>∈</sup> *<sup>X</sup> is reduced to IFS, denoted by <sup>A</sup>* <sup>=</sup> { *<sup>x</sup>*, *<sup>μ</sup>A*(*x*), *<sup>ν</sup>A*(*x*)]∨*<sup>x</sup>* <sup>∈</sup> *<sup>X</sup>*}*, where <sup>μ</sup>A*(*x*) = [*μ<sup>l</sup> <sup>A</sup>*(*x*), *<sup>μ</sup><sup>r</sup> <sup>A</sup>*(*x*)] *and νA*(*x*) = *νl <sup>A</sup>*(*x*), *<sup>ν</sup><sup>r</sup> <sup>A</sup>*(*x*) *. Hence, Atanassov's IFS can be considered as a special case of IVIFS.*

*For a fixed <sup>x</sup>* <sup>∈</sup> *<sup>X</sup>*, *an object μ<sup>l</sup> <sup>A</sup>*(*x*), *<sup>μ</sup><sup>r</sup> <sup>A</sup>*(*x*) , *νl <sup>A</sup>*(*x*), *<sup>ν</sup><sup>r</sup> <sup>A</sup>*(*x*) *is called interval-valued intuitionistic fuzzy number (IVIFN). Let β* = ([*p*, *q*], [*r*,*s*]) *be an IVIFN, where* 0 ≤ *p* ≤ *q* ≤ 1, 0 ≤ *r* ≤ *s* ≤ 1 *and q* + *s* ≤ 1*. Then the score function [34,35] S of β is defined as S*(*β*) = (*p* − *r*) + (*q* − *s*)/2*, where S*(*β*) ∈ [0, 1]*. The accuracy function [34,35] H of β is defined as H*(*β*) = (*p* + *r*) + (*q* + *s*)/2, *where H*(*β*) ∈ [0, 1].

*Xu and Jian [35] compared two IVIFNs using score and accuracy functions which is defined below. Let β*<sup>1</sup> = ([*p*1, *q*1], [*r*1,*s*1]) *and β*<sup>2</sup> = ([*p*2, *q*2], [*r*2,*s*2]) *be two IVIFNs, then*

	- *If H*(*β*1) = *H*(*β*2)*, then β*<sup>1</sup> = *β*2;
	- *If H*(*β*1) < *H*(*β*2)*, then β*<sup>1</sup> < *β*2;

*Xu and Chen [36] proposed a similarity measure between two IVIFNs β<sup>1</sup> = ([p1, q1], [r1, s1]) and β<sup>2</sup> = ([p2, q2], [r2, s2]) defined as*

$$S(\beta\_1, \beta\_2) = \frac{1}{4}(|\mathbf{p}\_1 - \mathbf{p}\_2| + |\mathbf{q}\_1 - \mathbf{q}\_2| + |\mathbf{r}\_1 - \mathbf{r}\_2| + |\mathbf{s}\_1 - \mathbf{s}\_2|) \tag{2}$$

*Let β = ([p,q],[r,s]), β<sup>1</sup> = ([p1,q1],[r1,s1]), β<sup>2</sup> = ([p2,q2],[r2,s2]) be three IVIFNs and λ > 0. Some of their basic operational laws [34,37] are given below.*


**Definition 2.** *Let β<sup>j</sup> = ([pj, qj],[rj, sj]) (j = 1,2,* ... *,n) be a collection of IVIFNs, and w = (w1,w2,* ... *,wn) <sup>T</sup> be their associated weight vector, with 0* <sup>≤</sup> *wj* <sup>≤</sup> *1 and* <sup>∑</sup>*<sup>n</sup> <sup>j</sup>*=<sup>1</sup> *wj* = 1, *then he interval valued intuitionistic fuzzy weighted geometric (IVIFWG) operator is defined as*

$$\text{IVIFWG}(\beta\_1, \beta\_2, \dots, \beta\_n) = \prod\_{j=1}^n \beta\_j^{\text{w}\_j} = \left( \left[ \prod\_{j=1}^n \mathbf{p}\_j^{\text{w}\_j}, \prod\_{j=1}^n \mathbf{q}\_j^{\text{w}\_j} \right], \left[ \prod\_{j=1}^n (1 - \mathbf{r}\_j)^{\text{w}\_j}, \prod\_{j=1}^n (1 - \mathbf{s}\_j)^{\text{w}\_j} \right] \right) \tag{3}$$

**Definition 3.** *According to Park et al. [38] the distance measures between two IVIFSs are defined as follows:*

*The Hamming distance dH*(*β*1, *β*2) *and Euclidean distance dE*(*β*1, *β*2) *for the IVIFNs β<sup>1</sup> = ([p1, q1], [r1, s1]) and β<sup>2</sup> = ([p2, q2], [r2, s2]) are computed as:*

$$\mathbf{d}\_{\rm H}(\beta\_1, \beta\_2) = \frac{1}{4}(|\mathbf{p}\_1 - \mathbf{p}\_2| + |\mathbf{q}\_1 - \mathbf{q}\_2| + |\mathbf{r}\_1 - \mathbf{r}\_2| + |\mathbf{s}\_1 - \mathbf{s}\_2|) \tag{4}$$

$$\mathrm{ch}\_{\mathrm{E}}(\boldsymbol{\beta}\_{1}, \boldsymbol{\beta}\_{2}) = \sqrt{\frac{1}{4} \left[ \left( \mathbf{p}\_{1} - \mathbf{p}\_{2} \right)^{2} + \left( \mathbf{q}\_{1} - \mathbf{q}\_{2} \right)^{2} + \left( \mathbf{r}\_{1} - \mathbf{r}\_{2} \right)^{2} + \left( \mathbf{s}\_{1} - \mathbf{s}\_{2} \right)^{2} \right]} \tag{5}$$

**Definition 4.** *Let <sup>A</sup>*<sup>1</sup> <sup>=</sup> *p* (1) *<sup>j</sup>* , *q* (1) *j* , *r* (1) *<sup>j</sup>* , *s* (1) *j* <sup>1</sup>×*<sup>n</sup> and <sup>A</sup>*<sup>2</sup> <sup>=</sup> *p* (2) *<sup>j</sup>* , *q* (2) *j* , *r* (2) *<sup>j</sup>* , *s* (2) *j* <sup>1</sup>×*<sup>n</sup> be two IVIFNs in the universe X = {x1, x2,* ... *, xn }, then the distance measure between β*<sup>1</sup> *and β*<sup>2</sup> *is defined as follows:*

$$\mathrm{D}\left(\check{\mathbf{A}}\_{1},\check{\mathbf{A}}\_{2}\right) = \left[\frac{1}{4\pi}\sum\_{j=1}^{n}\left\{ \left|\mathbf{p}\_{\uparrow}^{(1)} - \left|\mathbf{p}\_{\downarrow}^{(2)}\right|^{\lambda} + \left|\mathbf{q}\_{\downarrow}^{(1)} - \mathbf{q}\_{\downarrow}^{(2)}\right|^{\lambda} + \left|r\_{\uparrow}^{(1)} - r\_{\uparrow}^{(2)}\right|^{\lambda} + \left|s\_{\uparrow}^{(1)} - s\_{\uparrow}^{(2)}\right|^{\lambda} \right\} \right]^{1/\lambda} \tag{6}$$

*Particularly, if λ = 1, then Equation (6) becomes the Hamming distance:*

$$\mathbf{d}\_{\mathbf{H}}\{\tilde{\mathbf{A}}\_{1},\tilde{\mathbf{A}}\_{2}\} = \frac{1}{4n} \sum\_{j=1}^{n} \left\{ \left| \mathbf{p}\_{\mathbf{j}}^{(1)} - \mathbf{p}\_{\mathbf{j}}^{(2)} \right| + \left| \mathbf{q}\_{\mathbf{j}}^{(1)} - \mathbf{q}\_{\mathbf{j}}^{(2)} \right| + \left| \mathbf{r}\_{\mathbf{j}}^{(1)} - \mathbf{r}\_{\mathbf{j}}^{(2)} \right| + \left| \mathbf{s}\_{\mathbf{j}}^{(1)} - \mathbf{s}\_{\mathbf{j}}^{(2)} \right| \right\} \tag{7}$$

*If λ = 2, then Equation (6) is degenerated to the Euclidean distance:*

$$\mathbf{d}\_{\mathbf{H}}(\widetilde{\mathbf{A}}\_{1}, \widetilde{\mathbf{A}}\_{2}) = \sqrt{\frac{1}{4n} \sum\_{j=1}^{n} \left\{ \left| \mathbf{p}\_{\mathbf{j}}^{(1)} - \mathbf{p}\_{\mathbf{j}}^{(2)} \right|^{2} + \left| \mathbf{q}\_{\mathbf{j}}^{(1)} - \mathbf{q}\_{\mathbf{j}}^{(2)} \right|^{2} + \left| \mathbf{r}\_{\mathbf{j}}^{(1)} - \mathbf{r}\_{\mathbf{j}}^{(2)} \right|^{2} + \left| \mathbf{s}\_{\mathbf{j}}^{(1)} - \mathbf{s}\_{\mathbf{j}}^{(2)} \right|^{2} \right\} \tag{8}$$

#### **4. Proposed CODAS Method Using IVIFNs**

This section presents an extension of the CODAS method based on IVIFS to deal with MCDM problems. Let D = {d1, d2, ... , dl} be the group of decision makers, C = {c1, c2, ... , cn} be the set of criteria, and A = {a1,a2, ... ,am} be the set of alternatives. The group of experts/decision makers D = {d1, d2, ... , dl} provide their opinions regarding the criteria C = {c1, c2, ... , cn} corresponding to each alternative A = {a1, a2, ... , am} using linguistic terms which are presented by IVIFNs. In this algorithm, we consider that significance of individual decision makers are different and the weights of the decision makers are expressed using fuzzy membership grades. We also consider that opinions of individual decision makers about the importance of various criteria are different. A flow chart of the proposed approach is given in Figure 1.

**Figure 1.** Flow diagram of the proposed IVIF-CODAS.

A step wise illustration of the proposed approach is given below. **Step 1.** Opinion of each expert is expressed using decision matrix given below.

$$\mathbf{x}^{\mathbf{k}} = \begin{pmatrix} \mathbf{x}\_{11}^{\mathbf{k}} & \mathbf{x}\_{12}^{\mathbf{k}} & \dots & \mathbf{x}\_{1n}^{\mathbf{k}}\\ \mathbf{x}\_{21}^{\mathbf{k}} & \mathbf{x}\_{22}^{\mathbf{k}} & \dots & \mathbf{x}\_{2n}^{\mathbf{k}}\\ \dots & \dots & \dots & \dots & \dots\\ \mathbf{x}\_{m1}^{\mathbf{k}} & \mathbf{x}\_{m2}^{\mathbf{k}} & \dots & \mathbf{x}\_{mn}^{\mathbf{k}} \end{pmatrix} \quad (\mathbf{k} = 1, 2, \dots, \mathbf{l})\tag{9}$$

Here x<sup>k</sup> ij denotes the evaluating value of *i*th (i ∈ {1, 2, . . . , m}) alternative with respect to *j*th (j ∈ {1, 2, . . . , n}) criterion and *k*th (k ∈ {1, 2, . . . , l}) decision maker which is expressed as IVIFNs.

**Step 2.** Interval-valued intuitionistic fuzzy weighted geometric (IVIFWG) aggregation operator is used to aggregate the opinion of individual decision makers. The aggregated/collective decision matrix is formed as

$$\mathbf{X} = \begin{pmatrix} \mathbf{x}\_{11} & \mathbf{x}\_{12} & & \dots & \mathbf{x}\_{1n} \\ & \mathbf{x}\_{21} & \mathbf{x}\_{22} & & \dots & \mathbf{x}\_{2n} \\ & \dots & \dots & \dots & \dots \\ & \mathbf{x}\_{m1} & \mathbf{x}\_{m2} & \dots & \dots & \mathbf{x}\_{mn} \end{pmatrix} \tag{10}$$

where xij <sup>=</sup> IVIFWG x1 ij, x2 ij , ...,x<sup>l</sup> ij and l be the number of decision makers.

**Step 3.** Calculate the weights of evaluation criteria

Let w = (w1, w2, ...,wn) <sup>T</sup> be the weight vector of the criteria Cj, j = 1, 2, ... , n, where wj > <sup>0</sup> (∀j) and <sup>∑</sup><sup>n</sup> <sup>j</sup>=<sup>1</sup> wj = 1. The known criteria weights are divided into five basic ranking forms [39–41] for i = j as given below:


For simplicity, let H denote the set of criteria weight information given by DMs and *H* = *H*<sup>1</sup> ∪ *H*<sup>2</sup> ∪ *H*<sup>3</sup> ∪ *H*<sup>4</sup> ∪ *H*5. This approach uses the known criteria weight information to define the weights of evaluation criteria.

In MSP, the significance of a criterion is determined by evaluating the performance values of the alternatives for that criterion. When the performance values of the alternatives differ a little for a particular criterion, then that criterion is considered to be less significant for choosing the best material. Similarly, when the performance values of the alternatives differ much for a particular criterion, then that criterion is considered to be much significant for choosing the best material. This observation leads to assign less weight to the less significant criteria and more weight to the much more significant criteria [42]. A criterion is said to have no significance in the material selection process when the performance values of the alternatives are same for that criteria [43].

When the criteria weight information is partially known, this paper presents an optimization model using the IVIF distance measure to compute the evaluating criteria weights. Below, we define the distance between the alternative *Ai* and other alternatives corresponding to the criterion *Cj*.

$$\mathbf{D}\_{\overline{\mathbf{i}}} = \frac{1}{\mathbf{m} - 1} \sum\_{\mathbf{g} = 1, \mathbf{g} \neq \mathbf{i}}^{\mathbf{m}} \mathbf{d}\_{\overline{\mathbf{i}} \overline{\mathbf{i}}} (\mathbf{x}\_{\overline{\mathbf{i}} \overline{\mathbf{j}}}, \mathbf{x}\_{\overline{\mathbf{g}} \overline{\mathbf{i}}}); \mathbf{i} = 1, \ 2, \ \cdots, \mathbf{m}; \mathbf{j} = 1, \ 2, \ \cdots, \mathbf{m}. \tag{11}$$

The overall distance measures of all the alternatives for the criterion *Cj* is presented as:

$$\mathbf{D}\_{\mathbf{j}} = \frac{1}{\mathbf{m} - 1} \sum\_{\mathbf{i} = 1}^{\mathbf{m}} \sum\_{\mathbf{g} = 1, \mathbf{g} \neq \mathbf{i}}^{\mathbf{m}} \mathbf{d}\_{\mathbf{H}}(\mathbf{x}\_{\mathbf{i}|\mathbf{j}}, \mathbf{x}\_{\mathbf{g}|\mathbf{j}}), \mathbf{j} = 1, \ 2, \ \mathbf{\cdot} \cdot \mathbf{\cdot} \text{ , } \mathbf{n}. \tag{12}$$

Next the weighted distance function is formulated as given below.

$$\mathbf{D(w)} = \sum\_{\mathbf{j}=1}^{\mathbf{n}} \mathbf{D}\_{\mathbf{j}} \mathbf{w}\_{\mathbf{j}} = \sum\_{\mathbf{j}=1}^{\mathbf{n}} \sum\_{\mathbf{i}=1}^{\mathbf{m}} \mathbf{D}\_{\mathbf{i}\mathbf{j}} \mathbf{w}\_{\mathbf{j}} = \frac{1}{\mathbf{m}-1} \sum\_{\mathbf{j}=1}^{\mathbf{n}} \sum\_{\mathbf{i}=1}^{\mathbf{m}} \sum\_{\mathbf{g}=1,\mathbf{g}\neq\mathbf{i}}^{\mathbf{m}} \mathbf{d}\_{\mathbf{H}} (\mathbf{x}\_{\mathbf{i}\mathbf{j}}, \mathbf{x}\_{\mathbf{g}\mathbf{j}}) \mathbf{w}\_{\mathbf{j}} \tag{13}$$

Hence, a suitable weight vector of criteria *w* = (*w*1, *w*2,..., *wn*) *<sup>T</sup>* is needed to maximize *D*(*w*), and therefore, we can present the optimization model defined below:

$$\mathbf{P}(\mathbf{M} - 1) = \begin{cases} \text{Maximize } \mathbf{D}(\mathbf{w}) = \frac{1}{\mathbf{m} - 1} \sum\_{\mathbf{l} = 1}^{\mathbf{m}} \sum\_{\mathbf{l} = 1}^{\mathbf{m}} \sum\_{\mathbf{g} = 1, \ g \neq 1}^{\mathbf{m}} \mathbf{d}\_{\mathbf{fl}}(\mathbf{x\_{\mathbf{i}j}}, \mathbf{x\_{\mathbf{g}j}}) \mathbf{w}\_{\mathbf{j}} \\ \text{subject to } \mathbf{w} \in \mathbf{H}, \sum\_{\mathbf{l} = 1}^{\mathbf{m}} \mathbf{w}\_{\mathbf{j}} = 1, \quad \mathbf{w}\_{\mathbf{j}} \ge 0, \ j = 1, 2, \dots, n \end{cases} \tag{14}$$

The optimal solution w<sup>∗</sup> is obtained by solving the model (M − 1). We use the optimal solution w∗ as the weight vector for the evaluation criteria.

In another case, when the information concerning criteria weights is totally unknown, we can develop another model for optimization to find the optimal weights of criteria:

$$\mathbf{D}(\mathbf{M} - 2) = \begin{cases} \text{Maximize D(w)} = \frac{1}{\mathbf{m} - 1} \sum\_{\mathbf{l} = 1}^{\mathbf{m}} \sum\_{\mathbf{l} = 1}^{\mathbf{m}} \sum\_{\mathbf{g} = 1, \ g \neq 1}^{\mathbf{m}} \mathbf{d}\_{\mathbf{l}} (\mathbf{x}\_{\mathbf{l}\mathbf{y}}, \mathbf{x}\_{\mathbf{g}\mathbf{j}}) \mathbf{w}\_{\mathbf{j}} \\ \quad \text{subject to} \quad \sum\_{\mathbf{l} = 1}^{\mathbf{n}} \mathbf{w}\_{\mathbf{j}} = 1, \quad \mathbf{w}\_{\mathbf{l}} \ge 0, \ j = 1, 2, \dots, n \end{cases} \tag{15}$$

Lagrange's method is used to solve the preceding Model (15) and the corresponding optimal solutions are normalized to determine the criteria weight vector.

$$\mathbf{w}\_{\mathbf{j}} = \frac{\sum\_{\mathbf{i}=1}^{\mathbf{m}} \sum\_{\mathbf{g}=1, \mathbf{g} \neq \mathbf{i}}^{\mathbf{m}} \mathbf{d}\_{\mathbf{i}\mathbf{f}} (\mathbf{x}\_{\mathbf{i}\mathbf{j}}, \mathbf{x}\_{\mathbf{g}\mathbf{j}}) \mathbf{w}\_{\mathbf{j}}}{\sum\_{\mathbf{j}=1}^{\mathbf{m}} \sum\_{\mathbf{i}=1}^{\mathbf{m}} \sum\_{\mathbf{g}=1, \mathbf{g} \neq \mathbf{i}}^{\mathbf{m}} \mathbf{d}\_{\mathbf{i}\mathbf{f}} (\mathbf{x}\_{\mathbf{i}\mathbf{j}}, \mathbf{x}\_{\mathbf{g}\mathbf{j}}) \mathbf{w}\_{\mathbf{j}}} \tag{16}$$

**Step 4.** The collective decision matrix is normalized by determining the highest IVIFN under each criterion for all the alternatives and then performing the division operation of between the highest IVIFN and the corresponding IVIFN as given below.

$$\dot{\mathbf{N}} = \begin{bmatrix} \tilde{\mathbf{n}}\_{\ddot{\mathbf{j}}} \end{bmatrix}\_{\mathbf{m} \times \mathbf{n}} \tag{17}$$

where

$$\hat{\mathbf{m}}\_{\vec{\mathbb{M}}} = \left\{ \begin{array}{c} \frac{\mathbf{x}\_{\vec{\mathbb{M}}}}{\max\limits\_{1 \le i \le \mathbf{m} \atop 1 \le i \le \mathbf{m}}}; & \text{if } \mathbf{j} \in \mathbf{B} \\\ 1 - \frac{\max\limits\_{\mathbf{x}\_{\vec{\mathbb{M}}}}}{\max\limits\_{1 \le i \le \mathbf{m}} \left(\mathbf{x}\_{\vec{\mathbb{M}}}\right)}; & \text{if } \mathbf{j} \in \mathbf{C} \end{array} \right. \tag{18}$$

In the above operations, <sup>B</sup> and <sup>C</sup> respectively represent the sets of benefit and cost criteria, and <sup>n</sup>ij denote the normalized performance values in terms of IVIFNs.

**Step 5.** The weighted normalized decision matrix is determined by performing product operation between the aggregated criteria weights and the normalized performance values of the criteria corresponding to the alternatives, which is give below. It is noted that both the aggregated criteria weights and the evaluating values are expressed as IVIFNs.

$$
\dot{\mathbf{R}} = \begin{bmatrix} \check{\mathbf{r}}\_{\ddot{\mathbf{i}}} \end{bmatrix}\_{\mathbf{m} \times \mathbf{n}} \tag{19}
$$

where

$$
\widetilde{\mathbf{r}}\_{\overline{\mathbf{i}}\overline{\mathbf{j}}} = \mathbf{w}\_{\overline{\mathbf{j}}} \widetilde{\mathbf{n}}\_{\overline{\mathbf{i}}\overline{\mathbf{j}}} \tag{20}
$$

Here the weight of *j*th criterion is denoted by wj.

**Step 6.** The interval-valued intuitionistic fuzzy negative ideal solution is computed as

$$\mathbf{N}\overline{\mathbf{S}} = \left[\mathbf{\tilde{n}}\mathbf{\tilde{s}}\_{\mathbf{\tilde{i}}}\right]\_{\mathbf{m}\times\mathbf{n}}\tag{21}$$

where ns ij <sup>=</sup> min 1≤i≤m {rij}. Here ns ij is an IVIFN for each criterion j <sup>∈</sup> {1, 2, . . . , n}.

**Step 7.** IVIFN-based Hamming (HD) and Euclidean distances (ED) of the alternatives <sup>i</sup> <sup>∈</sup> {1, 2, . . . , m} from the interval-valued intuitionistic fuzzy negative ideal solution (NS, ) are computed as

$$\mathbf{i}\to\mathbf{D}=\sum\_{\mathbf{j}=1}^{n}\mathbf{d}\_{\mathbb{E}}\left(\widetilde{\mathbf{r}}\_{\mathbb{i}\mathbb{j}}\ \widetilde{\mathbf{r}}\mathbf{s}\_{\mathbb{i}\mathbb{j}}\right)\tag{22}$$

$$\text{HD} = \sum\_{\mathbf{j}=1}^{n} \mathbf{d}\_{\text{IF}} \left( \mathbf{\tilde{r}}\_{\text{ij}} \cdot \mathbf{n} \mathbf{\breve{s}}\_{\text{ij}} \right) \tag{23}$$

**Step 8.** The relative assessment matrix (RA) is computed as

$$\mathbf{RA} = \left\lfloor \mathbf{p}\_{\mathrm{is}} \right\rfloor\_{\mathrm{m} \times \mathrm{m}} \tag{24}$$

where

$$\mathbf{p}\_{\rm is} = \left(\mathbf{ED\_{l}} - \mathbf{HD\_{s}}\right) + \left\{\psi \left(\mathbf{ED\_{l}} - \mathbf{ED\_{s}}\right) \times \left(\mathbf{HD\_{l}} - \mathbf{HD\_{s}}\right)\right\} \tag{25}$$

where i, s ∈ {1, 2, . . . , m} and ψ is a threshold function defined below.

$$\Psi(\mathbf{t}) = \begin{cases} \quad \text{1;} & \text{if } |\mathbf{t}| \ge \Theta \\ \quad 0; & \text{if } |\mathbf{t}| < \Theta \end{cases} \tag{26}$$

Decision maker can set the threshold parameter (*θ*). Here we consider *θ* = 0.02. **Step 9.** The assessment score (*ASi*) of each alternative is computed, which is given below.

$$\text{AS}\_{\text{i}} = \sum\_{\mathbf{s}=1}^{\text{m}} \mathbf{p}\_{\text{is}} \quad (\forall \mathbf{i} \in \{1, 2, \dots, m\}) \tag{27}$$

#### **5. Application of the IVIF–CODAS in MSPs**

#### *5.1. Illustrative Example*

Here, a MSP is illustrated to show the applicability and effectiveness of the proposed IVIF-CODAS approach. We have considered an automotive parts factory in India [7], where the factory want to find the best material for the automotive instrument panel. In the selection process, a group of five experts/decision makers (DM1, DM2, DM3, DM4, DM5) evaluates four materials/alternatives (A1, A2, A3, A4) based on the values of eight evaluation criteria (C1, C2, C3, C4, C5, C6, C7, C8) and finds the suitable alternative. The structure of the problem in given in Figure 2. The linguistic assessments (using Table 1) of the alternatives given by the decision makers are shown in Table 2. This example assumes that the group of five decision makers and the set of eight criteria have their individual weights/importance. The weights of decision makers and criteria are respectively expressed using fuzzy membership grades and IVIFNs.

**Table 1.** Linguistic terms and corresponding IVIFNs for evaluating materials.


**Figure 2.** Structure of material selection in automotive industry.


**Table 2.** Decision matrix with linguistic ratings.


known in advance. Hence, applying model (*M* − 1) and assuming that the criteria weights are partially known as follows: *H* = {*w*<sup>1</sup> ≤ 0.08, *w*<sup>1</sup> = *w*2, 0.10 ≤ *w*<sup>3</sup> ≤ 0.20, *w*<sup>4</sup> + *w*<sup>2</sup> ≥ 0.3, 0.13 ≤ *w*<sup>5</sup> ≤ 0.20, 0.12 ≤ *w*<sup>6</sup> ≤ 0.17, 0.12 ≤ *w*<sup>7</sup> ≤ 0.16, *w*<sup>8</sup> = *w*6, *w*<sup>5</sup> − *w*<sup>6</sup> ≥ 0.05, *w*<sup>5</sup> − *w*<sup>7</sup> ≥ 0.03, *wj* ≥ 0, *j* = 1, 2, ... , 8, ∑<sup>8</sup> *<sup>j</sup>*=<sup>1</sup> *wj* = 1}. To compute the criteria priorities, Equations (16)–(18) and model (M − 1) are used to develop the linear programming model given below:

> ⎧ ⎪⎨ ⎪⎩ maxD(w) = 0.450w1 + 0.590w2 + 0.550w3 + 0.421w4 +0.550w5 + 0.530w6 + 0.640w7 + 0.605w8 subject to to w ∈ H



**Table 3.** Collective decision matrix.


**Table 4.** Weighted normalized decision matrix.

**Table 5.** Negative ideal solutions.


**Table 6.** Euclidean and Hamming distance matrices.



**Table 8.** Comparison with other models.


#### *5.2. A Real Case of MSP in Sustainable Construction Projects*

This section presents a sustainable material selection problem for construction projects using the proposed IVIF-CODAS approach. Applicability and usefulness of the proposed model are validated through a real case study of a construction company. Initially, we sent our research proposals to 10 national builders, who are associated with construction, materials supply, and providing services mainly in India. Three out of ten companies provided positive responses to our proposal and our research team accomplished necessary groundwork on these three construction companies which are located in the "Durgapur-Asansol" division of West Bengal. We have had selected "*Maha Prabhu Buiders: A national company (name changed)*" for implementing our proposed research framework. The company has several operating units all over the country. The primary goal of this study is to provide wide-ranging results by incorporating the perceptions of all actors in the construction field since this research work is completely based on the decision makers' (refer to Table 9) judgments. Therefore, we integrated some major perspectives: clients, the Construction Company, manufacturers, consultants, and material suppliers.



The main reason for selecting "*Maha Prabhu Buiders*" over other two volunteering companies is because they arrange for all kind of services such as construction, consultants and architects, material suppliers, and so on. The central public works department (CPWD-2014) of India has been promoting the norms and guidelines of sustainable construction in India to increase the awareness in public, private sectors, and most importantly among the habitants. Following to the CPWD (2014) guidelines, this particular company has been striving in sustainable construction projects and most of all other activities. The general manager discussed with our team and informed the new trend in clients. The trend is not limited to embracing the concept of sustainable design of buildings but also in materials (bricks, mortar, concrete, cement, plasters, etc.). Hence, they recognized our research framework, and acknowledged the results that will classify which sustainable materials are most preferable. The consultant will endorse these results, and as a product manufacturer, the company can advertise that specific material with its sustainability indicating tags. On acceptance of the research proposal, we arranged a three-day workshop for the managing directors of the company, architects, and aforementioned clients who encourage sustainability issues in designing buildings as well as selecting materials, engineers and skilled professionals from the company. The problem goal and structure were explained to all the participants and, after several rounds of discussion, the materials selected to be evaluated are burnt clay bricks. Bricks are used substantially in almost every type of construction projects including bridges, housing, firms, hospitals, etc.

As soon as the usefulness of the proposed framework was revealed, our research team began the preliminary investigation on the sustainable alternatives of burnt clay bricks. We found there are three popular alternative bricks which are usually used in sustainable construction projects. The detailed description and characteristics of alternatives bricks can be found in Dhanjode et al. [44] and Mahendran et al. [45].

For evaluating the alternative bricks for sustainable practices, the team of experts (*DM*1, *DM*2, *DM*3, *DM*4, *DM*5) conducts the performance testing to decide the most suitable brick. Next, the

sustainable indicators are considered as evaluation criteria which were recommended by the team of experts of the company and approved with the literature. In this concern the proposed framework is applied to find the most sustainable brick(s) based on the considered criteria of sustainable practices in construction industry. To accomplish the purpose of this assignment, a three-phase methodology was used: data collection, data aggregation using IVIFWG operator, and evaluation of bricks and selecting the most suitable brick(s) using extended IVIF-CODAS method. The decision structure of the problem is depicted in Figure 3 in which the goal of this project is located in the top level, followed by the most cited factors and criteria of sustainability. Final level of the hierarchy deals with evaluation of the alternative bricks as sustainable material for building construction.

**Figure 3.** Structure of sustainable material selection problem in construction industry.

In the selection process, a group of five experts (*DM*1, *DM*2, *DM*3, *DM*4, *DM*5) evaluates three sustainable materials/bricks (*A*1, *A*2, *A*3) based on the values of twelve evaluation criteria (*C*1, *C*2, *C*3, *C*4, *C*5, *C*6, *C*7, *C*8, *C*9, *C*10, *C*11, *C*12) and finds the suitable material. Similar to the previous case study, the preferences of the decision makers about the materials against the criteria are given in linguistic terms as shown in Table 1. This example also assumes that the group of five decision makers and the set of twelve criteria have their individual weights/importance. The weights of decision makers and criteria are respectively expressed using fuzzy membership grades and IVIFNs. The structure of the sustainable material selection problem is given below in Figure 3.

#### **Step 1.** *Data collection*

After finalizing the hierarchical structure (Figure 3) of brick selection problem the required data are collected for conducting the evaluation process. For evaluating the sustainable material for construction projects, sustainability parameters are regarded as criteria in this study. In search of the most suitable sustainability parameters, we go through the present literatures available in reliable journals such as Taylor & Francis, Elsevier, MDPI, and Springer. Articles with environmental, economical, and social sustainability indicators and sustainable material selection and evaluation in the context of the construction and manufacturing projects were selected for this research work. After

discussion with the relevant professionals, some criteria were selected which are given in Table 10. Next, each decision maker opined on the importance of individual criterion. Finally, the alternatives (list of bricks) were designed, which was already completed in previous steps.



#### **Step 2.** *Calculation of criteria weights*

Similar to previous example, according to model (*M* − 1) and Equation (14), the weight vector of the three objectives/dimensions (economic, environmental, and social) is calculated as (*wEC*, *wEN*, *wSO*) *<sup>T</sup>* = (0.4268, 0.3568, 0.2164) *<sup>T</sup>*. Likewise, the local and global criteria weights are computed and presented in Table 11.


**Table 11.** Aggregated weights of sustainable material selection indicators.

#### **Step 3.** *Evaluating the best sustainable material using IVIF-CODAS method*

The IVIF-CODAS method is used to prioritize the optimal choice for sustainable material selection problem. We follow the step-by-step calculation mechanism is discussed in Section 4 and phase 3 in Figure 1. The mechanism includes fuzzy aggregation of individual judgments followed by normalization. The criteria weights that are obtained using Equations (11)–(16) in the second phase are lodged via Equation (20) to the normalized data yielding weighted normalized decision matrix. Next, we find the negative ideal solutions (NIS) are defined in Equation (21), which are used as criteria reference points to obtain optimal solution. The Euclidean (ED) and Hamming (HD) distances between the interval-valued intuitionistic fuzzy negative ideal solution and criteria functions in the weighted normalized decision matrix(R) are calculated using Equation (19). Finally, we determine the relative assessment matrix (RA) which is computed according to Equation (24). Now, assessment score of each alternative material is calculated taking the column sum of the RA matrix. Ultimately, the alternative/material with highest assessment score is the most desirable alternative in the material selection problem. In our problem, we find the "Fly ash bricks (*A*3)" is the optimal choice for sustainable construction projects in India.

#### **6. Result Discussion**

The goal of this section is to analyze the results of the proposed framework in order to validate its rationality and practicality. For exploring the most significant dimensions/aspects of construction projects and to avoid complexity, the sustainability indicators (criteria) were classified into 3 major aspects of sustainability. To keep it simple, we assume all three dimensions should be equally important. On contrast, each individual criterion has its own importance as a sustainable indicator for evaluating sustainable materials in construction industry. Table 11 demonstrates economic (*C*1, *C*2, *C*3, *C*4), environmental (*C*5, *C*6, *C*7, *C*8), and social (*C*9, *C*10, *C*11, *C*12) aspects including the corresponding criteria within them. From Table 11, it is clear that "Prospective recycling and reusing" (*C*5), "Returns" (*C*3), and "Use of local material" (*C*11) are top three important criteria among the twelve sustainable indicators considered here. However, usually there are conflicting concerns on priorities of the economic and environmental factors [18]. Hence, it becomes difficult to adjust them under such conflicts. However, in the viewpoint of sustainable construction in India, our findings show that "Returns" (*C*3) is the most important criterion in the economic dimension. It is quite reasonable for a company to devote its main effort to get higher returns. This fact is supported by the results we obtained by IVIFWG operator. In this cluster, the descending order of criteria is *C*3 > *C*1 > *C*2 > *C*4.

For the second dimension, "Prospective recycling and reusing" (*C*5) occupies the most significant sustainable indicator among the four environmental factors. In this group, the descending order of criteria is as follows: *C*5 > *C*6 > *C*8 > *C*7. Now, to justify such order of criteria importance, some of the works in the existing literature are cited. For green practices the 3R (reduce, recycle and reuse) policy [46,47] is the most sustainable strategy for providing the promotion of environmental performance and development. Finally, "health and safety" (*C*11) grips the top rank in the social dimension. Construction companies in a nation like India deals with more issues than other countries, since a large number of accidental cases as well as safety-related difficulties happen in construction. Table 11 reflects this fact that "Health and safety" (*C*11) and "Fire resistance" (*C*12) come first rather thinking much of "Labor availability" (*C*9), and "Esthetics" (*C*10) in real life situations.

After finalizing the criteria weights, we aim to evaluate and select the most suitable brick(s) as alternative material for sustainable construction. The linguistic assessments of the three alternatives supplied by the five decision makers are shown in Table 12. The decision matrix in Table 12 is transformed to the aggregated IVIF decision matrix (Table 13) by using Equation (10). Next, the average decision matrix is normalized with the help of Equation (18) and after plugging the criteria weights the weighted normalized decision matrix (refer to Table 14) is computed. Interval-valued intuitionistic fuzzy negative ideal solution for each of the criteria is shown in Table 15. Hamming distance and Euclidean distance of the alternatives from the Interval-valued intuitionistic fuzzy negative ideal solution is shown in Table 16. Finally, Table 17 tells the overall performance scores of all the sustainable alternatives of brick, in which "Fly ash bricks" (*A*3) bears the top rank with a high assessment score of 2.158. In addition, "AAC bricks" (*A*3) and "Clay Bricks" (*A*2) have got second and third ranks, with assessment scores −0.976 and −1.182, respectively. The final ranking order of alternative bricks for sustainable construction is: *A*3 > *A*1 > *A*2. Additionally, Table 18 shows the ranking orders by the proposed method along with four other existing methods for result comparisons.


**Table 12.** Decision matrix with linguistic ratings.

**Table 13.** Aggregated IVIF decision matrix.


On comparison with Clay bricks (*A*2), both of the Fly ash bricks (*A*3) and AAC bricks (*A*1), which consume less energy and have better thermal insulation, are found to be stronger than conventional clay bricks. In addition to this, although Dhanjode et al. [44] argued that the Fly ash bricks and AAC brick save 82 and 95 carbon tax respectively compared to clay bricks to its environmental properties. However, in Indian perspectives, only a few research works are found on AAC bricks while AAC bricks have been used for the construction of several buildings. Global unavailability of the AAC bricks often arises serious concern since it increases transportation and inventory costs. Moreover, Fly ash bricks are more cost and energy saving alternative choice than other ones. Mahendran et al. [45] asserted that Fly ash brick is the most favorable choice among the building blocks in perspectives of strength, heating load, framed and load bearing buildings. The above findings have selected Fly ash brick to be the most sustainable construction material for use in this case study. When we shared our results to the relevant builders, they appreciated our task and adopted this material in their own

construction business. The builders also discussed with their clients about the benefits of using this material and suggested them to use it.


**Table 14.** Weighted normalized decision matrix.

**Table 15.** Negative ideal solutions.


**Table 16.** Euclidean and Hamming distance matrices.






#### *6.1. Comparisons*

In this section, we perform the necessary comparative analysis with a set of a few existing approaches to prove the practicality and efficiency of the proposed IVIF-CODAS method. The CODAS [16], fuzzy CODAS [48], IVIF-TOPSIS [49], IVIF-VIKOR [50] methods were modified using fuzzy and IVIF numbers. The reason behind choosing these methods is their stability and reliability in producing satisfactory solutions/results. It is essential to measure the reliability of the results attained by the proposed model for crosschecking the optimal results/alternatives. For such action in MCDM problems the most common measure is the comparison of the results produced by other stable and reliable methods [16,51].

Now, the classical CODAS needs crisp numbers as inputs. Therefore, the crisp numbers corresponding to their linguistic ratings are used to perform the CODAS algorithm. Here, the order of ranking is similar to that of the original study. On contrast, fuzzy CODAS can adopt linguistic ratings as triangular/trapezoidal fuzzy numbers. The outcome of fuzzy CODAS is also same as the ranking produced by IVIF-CODAS. Finally, IVIF-TOPSIS and IVIF-MABAC are used to solve the same material selection problem. Table 18 shows that *A*3 (Fly ash bricks) occupies the first ranking in all the methods except in IVIF-VIKOR. There is a ranking swapping between *A*3 (Fly ash bricks) and *A*1 (AAC bricks).

From the comparative analysis, we can summarize that the results are harmonious to each other. Below we give some advantages of the proposed IVIF-CODAS method.


consider this to be one of the advantages of the novel approach that is reckoned to be applicable irrespective of its case studies.

#### *6.2. Sensitivity Analysis*

It is well known that the results of MCDM methods can be highly influenced by the weight coefficients of the evaluation criteria. Hence, a sensitivity analysis is discussed in the next two paragraphs, where we compute the final ranking of the alternative bricks by changing the criteria weights. Small variations in the rank of alternatives are noticed due to the small variations in the weight coefficients. Hence, the results of MCDM methods are analyzed by their sensitivity to these changes [55,56]. Below we present a sensitivity analysis using 10 different scenarios (Table 19). We apply the expression given in Equation (28) for changing the criteria weights.

$$w\_j^{new} = w\_j^{old} \pm \kappa w\_j^{old} \tag{28}$$


**Table 19.** Ranking for different scenarios.

Here *α* is the percentage of change of *wold <sup>j</sup>* . In this work, the sustainable material selection and criteria weights evaluation are performed by human inputs. The robustness testing of the final ranking has been conducted by considering the changed weights of the criteria. Small changes in criteria weights of the alternatives *A*1, *A*2 and *A*3 have a negligible effect in the final ranking in the brick selection problem. The result of the performed sensitivity analyses enforce the proposal that Fly ash bricks (*A*3) have the highest priority followed by AAC bricks (*A*1) and Clay bricks (*A*2). The computed ranking order (Figure 4) *A*3 (Fly ash bricks) *A*1 (AAC bricks) *A*2 (Clay bricks) can be followed in eight out of 10 scenarios. Thus, *A*3 (Fly ash bricks) has been ranked as maximum number of scenarios except in scenario 5 and 10. In these two cases, there are noticeable changes (increase and or decrease) in priorities of criteria set which obviously affect the final ranking as *A*1 (AAC bricks) *A*3 (Fly ash bricks) *A*2 (Clay bricks).

**Figure 4.** Different ranking positions due to sensitivity analysis.

We find the ranking is consistent unless some noticeable changes are made in the criteria weights. The robustness of the sustainable materials has been shown by sensitivity analysis (Table 19). *A*3 (Fly

ash bricks) or *A*1 (AAC bricks) occupies top rank in all cases and may be selected as best by the decision makers. The sensitivity analysis becomes meaningful to assess the bricks as sustainable building blocks of construction projects.

#### **7. Conclusions**

Sustainable material selection as well as suitable material selection in uncertainty for construction industry is mandatory to sustain in the competitive market and for eco-friendly environment. Since material selection is one kind of MCDM problem, an efficient MCDM method is needed to cope with the market challenges. In this study, the classical CODAS method is extended with IVIFNs known as IVIF-CODAS for comprehensive, rational, and sensible decision making and especially to handle the uncertain material selection problem. Since human reasoning capability is inherently inexact in nature, we have used linguistic terms to present the opinions of decision makers. IVIFN has been used in this study, since it uses intervals rather than exact numbers to represent the membership and non-membership functions. The comparative study shows that the proposed method is consistent and efficient in respect to the other existing methods. This ranking method will enable clients, the Construction Company, manufacturers, consultants, and material suppliers to understand central public works department norms in India and keep at parity with global standards.

Although this research article delivers numerous valuable implications, a few of the project implications are mentioned here. In any industry it is highly important to keep a balance between the cost parameters (material cost and its operating cost, etc.); hence, managers or engineers face difficulty in selecting the finest material only on the basis of its sustainability benefit. Also, the incorporation of TBL (environment, economy, and society) factors is equally difficult and needs a typical framework in material selection problem. This research work offers such a framework that aids the engineers and experts to decide the most suitable material under sustainability constraints especially in the Indian construction industries. Most research papers in the literature mainly focused on environmental concerns in the construction sector. In comparison to them this paper assists both scientific and societal contributions which contributes towards social progress along with the eco-friendly set-ups. Additionally, this study functions as a reference for researchers and practitioners working in the same field in an Indian context. As a whole, this paper helps and inspires architects, engineers, and other construction managers to respond and to decide the best sustainable material within their viable business set-up.

In future, IVIF-CODAS method can be used to solve other complex MCDM problems, like market segment evaluation and selection, project selection, supplier selection, etc.

**Author Contributions:** The individual contribution and responsibilities of the authors were as follows: J.R., S.D. and S.K. designed the research, collected and analyzed the data and the obtained results, performed the development of the paper. D.P. provided extensive advice throughout the study, regarding the research design, methodology, findings and revised the manuscript. All the authors have read and approved the final manuscript.

**Funding:** This work was supported by the Department of Science and Technology, India under the INSPIRE fellowship programme [grant no. DST/INSPIRE Fellowship/2013/544].

**Acknowledgments:** We wish to express our deepest appreciation to the editors and the anonymous reviewers.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **A New Hybrid MCDM Model: Sustainable Supplier Selection in a Construction Company**

#### **Bojan Mati´c 1, Stanislav Jovanovi´c 1, Dillip Kumar Das 2, Edmundas Kazimieras Zavadskas 3, Željko Stevi´c 4,\*, Siniša Sremac <sup>1</sup> and Milan Marinkovi´c <sup>1</sup>**


Received: 21 February 2019; Accepted: 3 March 2019; Published: 8 March 2019

**Abstract:** Sustainable development is one of the most important preconditions for preserving resources and balanced functioning of a complete supply chain in different areas. Taking into account the complexity of sustainable development and a supply chain, different decisions have to be made day-to-day, requiring the consideration of different parameters. One of the most important decisions in a sustainable supply chain is the selection of a sustainable supplier and, often the applied methodology is multi-criteria decision-making (MCDM). In this paper, a new hybrid MCDM model for evaluating and selecting suppliers in a sustainable supply chain for a construction company has been developed. The evaluation and selection of suppliers have been carried out on the basis of 21 criteria that belong to all aspects of sustainability. The determination of the weight values of criteria has been performed applying the full consistency method (FUCOM), while a new rough complex proportional assessment (COPRAS) method has been developed to evaluate the alternatives. The rough Dombi aggregator has been used for averaging in group decision-making while evaluating the significance of criteria and assessing the alternatives. The obtained results have been checked and confirmed using a sensitivity analysis that implies a four-phase procedure. In the first phase, the change of criteria weight was performed, while, in the second phase, rough additive ratio assessment (ARAS), rough weighted aggregated sum product assessment (WASPAS), rough simple additive weighting (SAW), and rough multi-attributive border approximation area comparison (MABAC) have been applied. The third phase involves changing the parameter *ρ* in the modeling of rough Dombi aggregator, and the fourth phase includes the calculation of Spearman's correlation coefficient (SCC) that shows a high correlation of ranks.

**Keywords:** sustainability; supplier selection; construction; FUCOM; rough COPRAS; rough Dombi aggregator

#### **1. Introduction**

Sustainable engineering implies the execution of all processes and activities respecting all aspects of sustainability: economic, social, and environmental aspects. In addition, it is necessary to take into account the interactions and symmetry between them. This is confirmed by Hutchins et al. [1] according to whom it is necessary to define and understand the relationships that exist among aspects of sustainability and how they influence each other. In the last two decades, according to Vanalle et al. [2], companies around the world have been showing increasing concern about the impact of their operations on the environment, which arises as a result of pressure by legal regulations, customers, and competitors. Taking this into account, construction companies operate under great pressure due to their potentially negative impact on the environment and a complete, sustainable supply chain. In line with sustainability—that has become inevitability—and urgent need, supply chains are also changing, and their focus is no longer just on rationalizing costs, but also on environmental concerns. On this basis, sustainable supply chain management (SSCM) and green supply chain management (GSCM) have been established. The SSCM concept, according to Sen et al. [3], is an integrated approach that links economic and social thinking together with environmental awareness in traditional supply chain management. SSCM is based on the idea that in addition to constant monitoring of economic values, companies must consider environmental and social aspects, too. This implies that, in order to achieve sustainability, companies should solve environmental issues together with meeting social standards at all levels of supply chain [4], and at the same time, achieving certain economic effects. In order to achieve the effects of SSCM, according to Rabbani et al. [5], a large number of individual participants in a supply chain, starting from suppliers to top managers, have to take into account sustainable aspects. At the very beginning of the sustainability concept, according to Singh and Trivedi [6], the focus was mainly on environmental issues, and much less on social aspects, as it was thought that by managing and reducing negative impacts on the environment, companies would achieve competitive advantages. Nowadays, it is a different situation and, therefore, the evaluation and selection of suppliers based on an equal number of criteria by all aspects of sustainability has been performed in this paper.

This paper has several interrelated objectives. The first aim of this research refers to the development and detailed description of the algorithm of a new rough complex proportional assessment (COPRAS) method. The second aim that appears as a causal link to the previous one refers to the development of a new hybrid model which implies the integration of full consistency method (FUCOM), rough Dombi aggregator, and rough COPRAS method. The third aim of the paper is to popularize the FUCOM method, which contributes to the objective determination of weight criteria values, as well as to popularize the application of MCDM methods in integration with rough numbers.

After the introductory part, which explains the aims and motivation for this research, the paper consists of five more sections. The second section presents a two-phase procedure for reviewing the situation in the field. A review of MCDM methods in sustainable civil engineering and a review of MCDM methods for sustainable supplier selection are presented. The third part includes the developed methodology of this paper. At the beginning of the section, the process of research is presented with the contributions and advantages of this paper. Then, the FUCOM method is briefly explained in the first part, while the algorithm of the developed rough COPRAS method is elaborated and explained, in detail, in the second part. The fourth section describes a complete procedure for the selection of a sustainable supplier in a construction company. A detailed calculation for each step of the developed methodology is presented in order to make the model much more understandable to readers. The fifth section is a sensitivity analysis and discussion. The sensitivity analysis implies the already described four-phase procedure, followed by a discussion of the results obtained. In the sixth section, concluding observations with paper contributions and guidelines for future research are provided.

#### **2. Literature Review**

#### *2.1. Review of MCDM Methods in Sustainable Civil Engineering*

Formal decision-making methods can be used to help improve the overall sustainability of industries and organizations [7]. According to Zavadskas et al. [8], as sustainable development is becoming more relevant, more and more articles are being published related to sustainability in the field of construction. According to same authors, sustainable decision-making in civil engineering, construction, and building technology can be supported by fundamental scientific achievements and multi-criteria decision-making (MCDM) theories that, according to Mardani et al. [9], are widely used. In the field of construction, increasing attention is being paid to energy efficiency and smart buildings, and therefore it is necessary to go towards sustainability in the design and construction of facilities and infrastructure.

Construction is an area that interacts enormously with the natural environment. A large percentage of raw materials are obtained from the earth, and in their treatment, processing, and the construction of buildings, certain environmental pollution is inevitable. Lombera and Rojo [10] use the Spanish MIVES (in English, integrated value model for sustainable assessment) methodology to define criteria for the sustainability of industrial buildings and to select the optimum solution with regard to them. A similar study is presented in [11], where authors also use the MIVES method but in combination with Monte Carlo simulation, in order to assess the sustainability of concrete structures. De la Fuente et al. [12] also apply the MIVES methodology together with the analytic hierarchy process (AHP) method in order to reduce subjective human impact on the selection of sewage pipe material. The MIVES methodology is also used in [13] in assessing the sustainability of alternatives—the types of concrete and their reinforcement for application in tunnels. The problem of monitoring, repairing, and returning to the function of steel bridge structures is a major challenge for engineers, especially because it is necessary to make key decisions, and wrongly made decisions can be very costly. In order to exclude subjectivity in selecting alternatives, Rashidi et al. [14] presented the decision support system (DSS), within which the simplified AHP (S-AHP) method was used. S-AHP combines simple multi-attribute rating technique (SMART) and AHP method. The aim is to help engineers in planning the safety, functionality, and sustainability of steel bridge structures. Jia et al. [15] present a framework for the selection of bridge construction between the ABC (Accelerated Bridge Construction) method and conventional alternatives, using the technique for order of preference by similarity to ideal solution (TOPSIS) and fuzzy TOPSIS methods.

Formisano and Mazzolani [16] present a new procedure for the selection of the optimum solution for seismic retrofitting of existing buildings which involves the application of three MCDM methods: TOPSIS, elimination and choice expressing reality (ELECTRE), and VlšeKriterijumska Optimizacija i Kompromisno Rešenje (VIKOR). Terracciano et al. [17] selected cold-formed thin-walled steel structures for vertical reinforcement and energy retrofitting systems of existing masonry constructions using TOPSIS method. Šiožinyte et al. [ ˙ 18] apply the AHP and TOPSIS grey MCDM methods to select an optimum solution for modernizing traditional buildings. Khoshnava et al. [19] apply MCDM methods to select energy efficient, ecological, recyclable materials for building, with respect to the three pillars of sustainability. In order to evaluate 23 criteria in the selection of materials, they use the decision-making trial and evaluation laboratory (DEMATEL) hybrid MCDM method together with the fuzzy analytic network process (FANP). Akadiri et al. [20] use fuzzy extended AHP (FEAHP) in order to select sustainable building materials. In [21], the ANP method is used to select an environmentally friendly method for the construction of a highway, since it can have a great impact on the environment. Most systems for evaluating the sustainability of facilities take into account only the environmental aspect and the environmental impact. However, it is necessary to take into account all three basic principles of sustainability, and thus Raslanas et al. [22], in their work, develop a system for evaluating the sustainability of recreational facilities using the AHP method. MCDM tool, according to Kumar et al. [23], is becoming popular in the field of energy planning due to the flexibility it provides to the decision-makers to take decisions while considering all the criteria and objectives simultaneously. MULTIMOORA and TOPSIS are used in [24] for sustainable decision-making in the energy planning. The authors have concluded that hydro and solar power systems were identified as the most sustainable. A study performed in [25] deals with developing a sustainability assessment framework for assessing technologies for the treatment of urban sewage sludge based on the logarithmic fuzzy preference programming-based fuzzy analytic hierarchy process (LFPPFAHP) and extension theory. Salabun et al. [26] developed an MCDM model with

COMET method for offshore wind farm localization. This method is also used in [27] for sustainable manufacturing and for solving the problem of the sustainable ammonium nitrate transport in [28].

#### *2.2. Review of MCDM Methods for Sustainable Supplier Selection*

The selection of suppliers is a constant process that requires the consideration of a certain number of criteria needed to make a decision on the selection of the most suitable suppliers [29–31]. According to Yazdani et al. [32], supplier evaluation and selection is a significant strategic decision for reducing operating costs and improving organizational competitiveness to develop business opportunities. Therefore, it is necessary to pay special attention to the selection of suppliers, including all aspects of sustainability.

The supplier selection, according to many authors, is one of the most demanding problems of sustainable supply chain management [33]. Fuzzy approach in combination with TOPSIS method is applied in [34] for assessing the sustainable performance of suppliers. In order to select suppliers in terms of sustainability, Dai and Blackhurst [35] present an integrated approach based on AHP and the quality function deployment (QFD) method. For the sustainable supplier selection, Azadnia et al. [36] propose an integrated approach that, in addition to the Fuzzy AHP method, is based on multi-objective mathematical programming, as well as on rule-based weighted fuzzy method. In [37], the assessment of sustainable supply chain management and the selection of suppliers are performed using grey theory in combination with the DEMATEL method, while Luthra et al. [38] present an integrated approach consisting of a combination of AHP and VIKOR method based on 22 criteria for all three aspects of sustainability. Sustainable supplier selection of raw materials in order to achieve sustainable development of the company is performed in [39], based on the fuzzy entropy–TOPSIS method. Hsu et al. [40] present a hybrid approach based on several MCDM methods in order to select suppliers in terms of carbon emissions. The evaluation of the supplier performance in the field of electronic industry in order to implement green supply chains is a topic of research in [41]. The authors use rough DEMATEL–ANP (R'AMATEL) in combination with rough multi-attribute ideal real comparative analysis (R'MAIRCA) method. Liu et al. [42] select the suppliers of fresh products using best worst method (BWM) and multi-objective optimization on the basis of the ratio analysis (MULTIMOORA) method. Kusi-Sarpong et al. [43] present a framework for ranking and selecting the criteria for sustainable innovations in supply chain management based on the BWM method. A quantitative assessment of the performance of a sustainable supply chain is presented in [44] based on fuzzy entropy and fuzzy Multi-Attribute Utility Theory (MAUT) methods. Das and Shaw [45] propose a model based on AHP and fuzzy TOPSIS method for selecting a sustainable supply chain, taking into account carbon emissions and various social factors. Luthra et al. [46] propose the application of Delphi and fuzzy DEMATEL methods for identifying and evaluating guidelines for the application information and communication technologies in sustainable initiatives in supply chains. In [47], a framework that identifies sustainable processes in supply chains for individual industries in India is presented. The ranking of industry branches is carried out using six fuzzy MCDM methods. Liou et al. [48] are proposed hybrid model consists of DEMATEL, ANP, and COPRAS-G methods for improving green supply chain management. They have used 12 criteria for supplier selection in the electronic industry, and provided a systemic analytical model for the improvement of parts of the supply chain management.

#### **3. Methods**

Figure 1 presents the methodology used in this paper, which consists of four phases:


**Figure 1.** Research flow with proposed model.

The first phase consists of five steps. First, recognition of the need for this research and definition of the problems and aims of the research are performed in the first two steps, and the MCDM model is formed in the third step. After forming the model and defining all elements, the criteria, the alternatives, and the team of experts, the processes of data collection begins, which is the fourth step of the first phase. In the last step, the evaluation of the mutual significance of the criteria and evaluation of the alternatives by the formed team of experts are carried out. The second phase consists of three steps, where the first one is collected data sorting and preparation for their insertion into the developed model. The second step is the development and detailed description of the rough COPRAS method, and the creation of a hybrid MCDM model in the last step of this phase.

The third phase provides a detailed calculation for the evaluation and selection of a sustainable supplier, which consists of four steps. First, the determination of the criteria values using the FUCOM method is carried out, and then the transformation of the obtained values into rough numbers, in order to perform averaging using the rough Dombi aggregator and obtain the final values of the criteria. Subsequently, in the third step, the rough Dombi aggregator is again used to obtain an initial rough matrix, in order to make a decision on the selection of a sustainable supplier using the rough COPRAS method in the fourth step. The final phase is the sensitivity analysis already explained in the previous section.

We have decided to extent the COPRAS method with rough numbers from following reasons. Rough set theory is vague, subjective, and imprecise, while the COPRAS method, according to Mulliner et al. [49], allows for both benefit and cost criteria to be incorporated with one analysis without difficulty or question. The main advantage of COPRAS method compared with other MCDM methods is to be able to show utility degree. Also, COPRAS method has a simple procedure to use.

The following is a brief summary of the FUCOM method algorithm in the first part and the detailed algorithm description of rough COPRAS method in the second part.

#### *3.1. Full Consistency Method (FUCOM)*

The FUCOM method has been developed by Pamuˇcar et al. [50] for determining the weights of criteria. It is a new method that, according to authors, represents a better method than AHP (analytical hierarchy process) and BWM (best worst method). So far, it has been applied in studies [51–54]. It consists of the following three steps:

Step 1. In the first step, the criteria from the predefined set of evaluation criteria *C* = {*C*1, *C*2,..., *Cn*} are ranked. The ranking is performed according to the significance of the criteria, i.e., starting from the criterion which is expected to have the highest weight coefficient to the criterion of the least significance.

$$\mathcal{C}\_{j(1)} > \mathcal{C}\_{j(2)} > \dots > \mathcal{C}\_{j(k)} \tag{1}$$

Step 2. In the second step, a comparison of the ranked criteria is carried out and the comparative priority (*ϕk*/(*k*+1), *k* = 1, 2, ... , *n*, where *k* represents the rank of the criteria) of the evaluation criteria is determined.

$$\Phi = \left( \varphi\_{1/2}, \varphi\_{2/3^\prime}, \dots, \varphi\_{k/(k+1)} \right) \tag{2}$$

Step 3. In the third step, the final values of the weight coefficients of the evaluation criteria (*w*1, *w*2,..., *wn*) *<sup>T</sup>* are calculated. The final values of the weight coefficients should satisfy the two conditions:

(1) that the ratio of the weight coefficients is equal to the comparative priority among the observed criteria (*ϕk*/(*k*+1)) defined in Step 2, i.e., that the following condition is met:

$$\frac{w\_k}{w\_{k+1}} = \varphi\_{k/(k+1)}.\tag{3}$$

(2) In addition to Condition (3), the final values of the weight coefficients should satisfy the condition of mathematical transitivity, i.e., that *ϕk*/(*k*+1) ⊗ *ϕ*(*k*+1)/(*k*+2) = *ϕk*/(*k*+2).

Since *<sup>ϕ</sup>k*/(*k*+1) <sup>=</sup> *wk wk*+<sup>1</sup> and *<sup>ϕ</sup>*(*k*+1)/(*k*+2) <sup>=</sup> *wk*+<sup>1</sup> *wk*+<sup>2</sup> , *wk wk*+<sup>1</sup> <sup>⊗</sup> *wk*+<sup>1</sup> *wk*+<sup>2</sup> <sup>=</sup> *wk wk*+<sup>2</sup> is obtained.

Thus, another condition that the final values of the weight coefficients of the evaluation criteria need to meet is obtained, namely

$$\frac{w\_k}{w\_{k+2}} = \varrho\_{k/(k+1)} \odot \varrho\_{(k+1)/(k+2)}.\tag{4}$$

Based on the defined settings, the final model for determining the final values of the weight coefficients of the evaluation criteria can be defined.

$$\begin{array}{ll}\min\chi & \text{s.t.}\\ \left|\frac{w\_{j(k)}}{w\_{j(k+1)}} - \varrho\_{k/(k+1)}\right| \leq \chi\_{\prime} & \forall j\\ \left|\frac{w\_{j(k)}}{w\_{j(k+2)}} - \varrho\_{k/(k+1)} \right| \odot \left|\varrho\_{(k+1)/(k+2)}\right| \leq \chi\_{\prime} & \forall j\\ \sum\_{j=1}^{n} w\_{j} = 1, \quad \forall j\\ w\_{j} \geq 0, \quad \forall j \end{array} \tag{5}$$

#### *3.2. A Novel Rough COPRAS Method*

The COPRAS method was expanded with rough numbers as part of a sensitivity analysis in the research [55]. So far, a complete algorithm that can enrich the theoretical field of multi-criteria decision-making has not been demonstrated. From this aspect, the algorithm presented below represents a significant contribution to the literature that addresses the problems of multi-criteria

decision-making. It should be pointed out that the COPRAS method with interval rough numbers has been developed in [56], which differs from the proposed algorithm in this paper.

Rough COPRAS consists of the following steps:

Step 1: Forming a multi-criteria model. In the initial step, it is necessary to create a multi-criteria model with all necessary elements. Create a set of *n* alternatives that will be evaluated based on *m* criteria assessed by *e* experts.

Step 2: Forming an initial matrix for group decision-making (6). In this step, it is necessary to transform the individual matrices formed by experts' evaluations into an initial group rough matrix. In order to achieve this, it is necessary to apply basic operations with rough numbers.

$$X = \begin{array}{c} \mathbb{C}\_1 \quad \mathbb{C}\_2 \quad \dots \quad \mathbb{C}\_m\\ X = \begin{bmatrix} A\_1 & \text{R}N(\mathbf{x}\_{11}) & \text{R}N(\mathbf{x}\_{12}) & \dots & \text{R}N(\mathbf{x}\_{1n})\\ \text{R}N(\mathbf{x}\_{21}) & \text{R}N(\mathbf{x}\_{22}) & \dots & \text{R}N(\mathbf{x}\_{2n})\\ \dots & \dots & \dots & \dots & \dots\\ \text{R}N(\mathbf{x}\_{m1}) & \text{R}N(\mathbf{x}\_{m2}) & \dots & \text{R}N(\mathbf{x}\_{mm}) \end{array} \tag{6}$$

where *RN*(*xij*) is an estimated value of the *i*th alternative in relation to the *j*th criterion, *n* is the number of alternatives, and *m* is the number of criteria.

Step 3: Normalization of the initial rough decision-making matrix applying the linear normalization procedure (7).

$$r\_{ij} = \frac{\mathbf{x}\_{ij}^L; \mathbf{x}\_{ij}^{II}}{\sum \mathbf{x}\_{ij}^I; \mathbf{x}\_{ij}^{II}} = \left[\frac{\mathbf{x}\_{ij}^L}{\sum \mathbf{x}\_{ij}^{II}}; \frac{\mathbf{x}\_{ij}^{II}}{\sum \mathbf{x}\_{ij}^L}\right] \tag{7}$$

Step 4: Forming a weighted normalized matrix using the following Formula (8):

$$D = \left[ d\_{ij}^{L}; d\_{ij}^{II} \right] = \left[ w\_{\!\!\!/}^{L} \times r\_{\!\!\!/ \!/ \!/ \!/} w\_{\!\!\!/}^{\!\!L} \times r\_{\!\!\!/ \!/}^{\!\!L} \right],\tag{8}$$

where *rL ij*;*r<sup>U</sup> ij* is the normalized rough value of the *i*th alternative in relation to the *j*th criterion and *wj* is the weight or significance of the *j*th criterion.

Step 5: In this step, it is necessary to calculate the sum of the weighted normalized values for both types of criteria, for benefit criteria using Equation (9):

$$\mathcal{S}\_{+i} = \begin{bmatrix} s\_{ij}^{+L}; s\_{ij}^{+\mathcal{U}} \end{bmatrix}\_{1 \times n'} \tag{9}$$

and for cost criteria using Equation (10):

$$\mathcal{S}\_{-i} = \begin{bmatrix} s\_{ij}^{-L}; s\_{ij}^{-II} \end{bmatrix}\_{1 \times n}. \tag{10}$$

Step 6: Determining the inverse summarized matrix for cost criteria (11):

$$\left(\left(S\_i^{-}\right)^{-1} = \left[\frac{1}{s\_{ij}^{-\overline{\cdot}\overline{\cdot}\overline{\cdot}}}; \frac{1}{s\_{ij}^{-\overline{\cdot}\overline{\cdot}}}\right].\tag{11}$$

Step 7: Determining the sum of the matrix for cost criteria (12) and the sum of its inverse matrix (13) so that two matrices 1 × 1 are obtained:

$$\mathbb{E}\left(\overline{\mathbf{S}\_i^{-}}\right) = \sum \left[ \mathbf{s}\_i^{-L}; \mathbf{s}\_i^{-II} \right]\_{1 \times 1'} \tag{12}$$

$$\left(\overline{S\_i^-}\right)^{-1} = \sum \left[\frac{1}{s\_{ij}^{-\mathrm{II}}}; \frac{1}{s\_{ij}^{-\mathrm{L}}}\right]\_{1 \times 1}.\tag{13}$$

Step 8: Determining the relative significance for each alternative. The relative weight *Qi* for the *i*th alternative is calculated applying Equation (14):

$$Q\_i = S\_{+i} \times \frac{\left(\overline{S\_i^-}\right)}{S\_{-i} \times \left(\overline{S\_i^-}\right)^{-1}}.\tag{14}$$

Step 9: Determining the priorities of alternatives. The priority in comparing the alternatives is identified on the basis of their relative weight, where the alternative with a higher relative weight value is given a higher priority or a rank, and the alternative with a maximum value represents the most acceptable alternative.

$$A^\* = \left\{ A\_i \vert \max\_i Q\_i \right\}.\tag{15}$$

#### **4. Case Study**

Sustainable supplier selection in the construction company was carried out on the basis of 21 criteria shown and explained in Table 1: economic, social, and environmental criteria. Each of these main criteria consists of seven subcriteria. The set of criteria used in this study was selected according to relevant literature, and based on interviews with authorized and managerial persons in the construction company. The first subcriterion that belongs to economic criteria C11 (costs/prices) and the sixth subcriterion (consumption of resources) are the cost criteria, while the others are the benefit criteria.




**Table 1.** *Cont.*

In this study, the team of five experts took part in the process of determination of weight coefficients of criteria and assessment of alternatives. Experts with a minimum of six years' experience in civil engineering were chosen. After interviewing the experts, the collected data were processed, and the aggregation of expert opinion was obtained. The collecting of data was carried out in the period from November 2018 until January 2019.

#### *4.1. Determining Criteria Weights Using the FUCOM Method*

In the following section, a detailed overview is provided of determining weight coefficients of the first-level criteria.

Step 1. In the first step, the decision-makers (DMs) ranked the criteria: DM1: C1 > C3 > C2; DM2: C1 > C2 > C3; DM3: C1 > C3 > C2; DM4: C3 > C1 > C2; and DM5: C1 > C3 > C2.

Step 2. In the second step, the decision-makers compared, in pairs, the ranked criteria from step 1. The comparison is made according to the first-ranked criterion, based on the above scale [1, 7]. This is how the importance of the criteria is obtained (*Cj*(*k*) ) for all the criteria ranked in step 1 (Table 2).


**Table 2.** Significance of criteria.

Based on the obtained significance of criteria, comparative significance values of criteria for each expert are calculated as follows:

> DM1 : *ϕC*1/*C*<sup>3</sup> = 2.2/1 = 2.2, *ϕC*3/*C*<sup>2</sup> = 2.8/2.2 = 1.27; DM2 : *ϕC*1/*C*<sup>2</sup> = 2.7/1 = 2.7, *ϕC*2/*C*<sup>3</sup> = 2.7/2.7 = 1; DM3 : *ϕC*1/*C*<sup>3</sup> = 3.1/1 = 3.1, *ϕC*3/*C*<sup>2</sup> = 3.4/3.1 = 1.10; DM4 : *ϕC*3/*C*<sup>1</sup> = 1.7/1 = 1.7, *ϕC*1/*C*<sup>2</sup> = 2/1.7 = 1.18; DM5 : *ϕC*1/*C*<sup>3</sup> = 1.6/1 = 1.6, *ϕC*3/*C*<sup>2</sup> = 1.9/1.6 = 1.19.

Step 3. Final values of weight coefficient should satisfy two conditions:

	- DM1 : *w*1/*w*<sup>3</sup> = 2.2, *w*3/*w*<sup>2</sup> = 1.27; DM2 : *w*1/*w*<sup>2</sup> = 2.7, *w*2/*w*<sup>3</sup> = 1; DM3 : *w*1/*w*<sup>3</sup> = 3.1, *w*3/*w*<sup>2</sup> = 1.10; DM4 : *w*3/*w*<sup>1</sup> = 1.7, *w*1/*w*<sup>2</sup> = 1.18; DM5 : *w*1/*w*<sup>3</sup> = 1.6, *w*3/*w*<sup>2</sup> = 1.19.

By applying Expression (5), the models for determining weight coefficients of the first-level criteria for each decision-maker can be defined:

*w*2 *<sup>w</sup>*<sup>3</sup> − 1 

 = *χ*,

 = *χ*,

 ,

$$\begin{array}{llll} \text{DM}\_{1}(\text{First level}) & \text{DM}\_{2}(\text{First level})\\ \min \chi\\ \text{s.t.} \begin{cases} \left| \frac{\overline{w\_{1}}}{\overline{w\_{3}}} - 2.2 \right| = \chi & \left| \frac{\overline{w\_{1}}}{\overline{w\_{2}}} - 1.27 \right| = \chi\_{\prime} \\ \left| \frac{\overline{w\_{1}}}{\overline{w\_{2}}} - 2.8 \right|, & \text{s.t.} \begin{cases} \left| \frac{\overline{w\_{1}}}{\overline{w\_{2}}} - 2.7 \right| = \chi\_{\prime} & \left| \frac{\overline{w\_{2}}}{\overline{w\_{3}}} - 1 \right| = \chi\_{\prime} \\ \left| \frac{\overline{w\_{1}}}{\overline{w\_{3}}} - 2.7 \right|, & \text{s.t.} \begin{cases} \left| \frac{\overline{w\_{1}}}{\overline{w\_{3}}} - 2.7 \right|, \\ \frac{\overline{w\_{1}}}{\overline{w\_{3}}} - 2.7 \right|, \\ \sum\_{j=1}^{3} w\_{j} = 1, & w\_{j} \ge 0, \end{cases} & \text{s.t.} \end{cases} \end{array}$$

$$\begin{array}{llll} \text{DM}\_3(\text{First level}) & \text{DM}\_4(\text{First level}) &\\ \min\chi\\ \text{s.t.} \begin{cases} \left| \frac{\overline{w\_1}}{\overline{w\_3}} - 3.1 \right| = \chi, & \left| \frac{\overline{w\_2}}{\overline{w\_2}} - 1.10 \right| = \chi, \\ \frac{\left| \frac{\overline{w\_1}}{\overline{w\_2}} - 3.4 \right|, & \text{s.t.} \end{cases} & \text{s.t.} \begin{cases} \left| \frac{\overline{w\_3}}{\overline{w\_1}} - 1.7 \right| = \chi, & \left| \frac{\overline{w\_1}}{\overline{w\_2}} - 1.18 \right| = \chi, \\ \frac{\left| \frac{\overline{w\_3}}{\overline{w\_2}} - 2 \right|, & \text{s.t.} \end{cases} & \text{M}\_2 \begin{cases} \left| \frac{\overline{w\_3}}{\overline{w\_1}} - 2 \right|, & \text{s.t.} \end{cases} \\ \end{cases}$$

DM5(First level) min*χ* 

$$\text{s.t.} \begin{cases} \left| \frac{\overline{w}\_1}{\overline{w}\_3} - 1.6 \right| = \chi\_{\prime} \quad \left| \frac{\overline{w}\_3}{\overline{w}\_2} - 1.19 \right| = \chi\_{\prime} \\ \left| \frac{\overline{w}\_1}{\overline{w}\_2} - 1.9 \right| \\ \quad \sum\_{j=1}^3 w\_j = 1, \qquad w\_j \ge 0, \quad \forall j \end{cases}$$

By solving the models presented, the values of weight coefficients for the first-level criteria for every decision-maker are obtained, as shown in Table 3.


**Table 3.** Values of weight coefficients for the first level of decision-making according to each DM.

The final values shown in the last column of Table 3 are obtained by rough operations and the rough Dombi aggregator. First, the transformation of individual matrices into a group rough matrix is performed as follows:

$$\widetilde{\varepsilon}\_{1} = \{0.552, 0.574, 0.618, 0.282, 0.465\},$$

*Lim*(0.552) <sup>=</sup> <sup>1</sup> 3 (0.552 <sup>+</sup> 0.282 <sup>+</sup> 0.465) <sup>=</sup> 0.433, *Lim*(0.552) <sup>=</sup> <sup>1</sup> 3 (0.552 + 0.574 + 0.618) = 0.581, *Lim*(0.574) <sup>=</sup> <sup>1</sup> 4 (0.552 <sup>+</sup> 0.574 <sup>+</sup> 0.282 <sup>+</sup> 0.465) <sup>=</sup> 0.468, *Lim*(0.574) <sup>=</sup> <sup>1</sup> 2 (0.574 + 0.618) = 0.596, *Lim*(0.618) <sup>=</sup> <sup>1</sup> 5 (0.552 + 0.574 + 0.618 + 0.282 + 0.465) = 0.498, *Lim*(0.618) = 0.618, *Lim*(0.282) <sup>=</sup> 0.282, *Lim*(0.282) <sup>=</sup> <sup>1</sup> 5 (0.552 + 0.574 + 0.618 + 0.282 + 0.465) = 0.498, *Lim*(0.465) <sup>=</sup> <sup>1</sup> 2 (0.282 <sup>+</sup> 0.465) <sup>=</sup> 0.373, *Lim*(0.465) <sup>=</sup> <sup>1</sup> 4 (0.552 + 0.574 + 0.618 + 0.465) = 0.552,

*Symmetry* **2019**, *11*, 353

$$\begin{array}{c} RN(c\_1^1) = [0.433, 0.581]; \ RNN(c\_1^2) = [0.468, 0.596]; \ RNN(c\_1^3) = [0.498, 0.618];\\ RN(c\_1^4) = [0.282, 0.498]; \ RNN(c\_1^5) = [0.373, 0.552].\end{array}$$

Subsequently, the rough Dombi aggregator is applied and final rough values of the criteria at the first decision-making level are obtained. The aggregation is performed as follows.

After the transformation has been completed, five rough matrices, to which the operations of the rough Dombi aggregator is applied, are obtained. As mentioned in the previous part of the paper, the research has involved five experts who are assigned the same weight values of 0.200. Based on the presented values, Expression (8) from [56], and assuming that *ρ* = 1 is at the position of C1, the aggregation of values is performed as follows:

$$RNDNGA(c\_1) = \begin{cases} \underline{\dim(c\_1)} = \frac{\sum\_{j=1}^{5} \underline{\dim(q\_j)}}{1 + \left\{\frac{5}{\alpha} \cdot w\left(\frac{1 - f\left(\underline{\dim(q\_j)}\right)}{f\left(\underline{\dim(q\_j)}\right)}\right)\right\}^{1/\delta}} = \frac{2.652}{1 + \left(0.200 \times \left(\frac{1 - 0.20}{0.20}\right) + 0.200 \times \left(\frac{0.202}{0.20}\right) + \dots + 0.200 \times \left(\frac{1 - 0.20}{0.20}\right)\right)} = 0.394 \times 10^{-4} \text{ m} \\\ \underline{\dim(c\_1)} = \frac{\sum\_{j=1}^{5} \underline{\dim(q\_j)}}{1 + \left\{\frac{5}{\alpha} \cdot w\left(\frac{1 - f\left(\underline{\dim(q\_j)}\right)}{f\left(\underline{\dim(q\_j)}\right)}\right)\right\}^{1/\delta}} = \frac{2.648}{1 + \left(0.200 \times \left(\frac{1 - f\left(\underline{\dim(q\_j)}\right)}{f\left(\underline{\dim(q\_j)}\right)}\right)\right) + 0.200 \times \left(\frac{1 - 0.20}{0.20}\right) + \dots + 0.200 \times \left(\frac{1 - 0.20}{0.20}\right)} = 0.567 \text{ m} \end{cases}$$

.

Similarly, the decision-makers have ranked the criteria of the second level and the significance of criteria is obtained (Table 4).


**Table 4.** The ranking and significance of the second-level criteria for a group of economic factors.

Based on the calculation, in the same way as with the criteria on the first level of decision-making, the calculation for the second decision-making level is made, and the values are shown in Table 5 for a group of economic criteria, in Tables 6 and 7 for a group of social criteria, and in Tables 8 and 9 for a group of environmental criteria.


**Table 5.** Values of weight coefficients for the second decision-making level according to each decision-maker for a group of economic criteria.



**Table 7.** Values of weight coefficients for the second decision-making level according to each decision-maker for a group of social criteria.



**Table 8.** The ranking and significance of the second-level criteria for a group of environmental factors.

**Table 9.** Values of weight coefficients for the second decision-making level according to each decision-maker for a group of environmental criteria.


Based on the significance of the groups of criteria (economic, social, and environmental) and applying Equation (5), the models for each decision-maker are formed. Solving these models, we obtain the values of weight coefficients per decision-makers (Table 10).

**Table 10.** Final values of criteria.



**Table 10.** *Cont.*

The final values of weight coefficients by all criteria are obtained by multiplying the weight coefficients of the main criteria with the subcriteria of the group to which they belong. As can be seen from Table 10, the most important criteria belong to the group of economic and then environmental criteria, which is understandable with regard to the area of existence of the company in which the research has been carried out.

#### *4.2. Ranking Alternatives Using a New Rough COPRAS Method*

Table 11 presents the evaluation of alternatives according to all criteria based on the linguistic scale 1–7. In evaluating the alternatives, five decision-makers participated, whose expertise has already been described in the previous section.


**Table 11.** Comparison of alternatives by five decision-makers.

*Symmetry* **2019**, *11*, 353

In order to be able to apply the developed methodology, the transformation of individual matrices into a group rough matrix is performed first. An example of calculating the value of the third alternative according to criterion C11 is given below:

$$
\widetilde{c}\_{11} = \{3, 3, 1, 2, 2\}\_{\prime\prime}
$$

*Lim*(3) = <sup>1</sup> (<sup>3</sup> <sup>+</sup> <sup>3</sup> <sup>+</sup> <sup>1</sup> <sup>+</sup> <sup>2</sup> <sup>+</sup> <sup>2</sup>) <sup>=</sup> 2.2, *Lim*(3) <sup>=</sup> 3, *Lim*(1) <sup>=</sup> 1, *Lim*(1) <sup>=</sup> <sup>1</sup> (3 + 3 + 1 + 2 + 2) = 2.2, *Lim*(2) = <sup>1</sup> (<sup>1</sup> <sup>+</sup> <sup>2</sup> <sup>+</sup> <sup>2</sup>) <sup>=</sup> 1.67, *Lim*(2) <sup>=</sup> <sup>1</sup> (3 + 3 + 2 + 2) = 2.5

$$\text{RN}\left(A\_3^1\right) = \text{RN}\left(A\_3^2\right) = [2.2, 3]; \text{ RN}\left(A\_3^3\right) = [1, 2.2]; \text{ RN}\left(A\_3^4\right) = \text{RN}\left(A\_3^5\right) = [1.67, 2.5].$$

Subsequently, the rough Dombi aggregator is applied and the final rough values of alternatives are obtained. The aggregation on the same example of the third alternative for criterion C11 is carried out as follows:

*RNDWGA*(*A*3) = ⎧ ⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎩ *Lim*(*A*3) <sup>=</sup> <sup>∑</sup><sup>5</sup> *<sup>j</sup>*=<sup>1</sup> *Lim*(*ϕj*) 1+ 5 ∑ *j*=1 *wj* <sup>1</sup>−*<sup>f</sup>*(*Lim*(*<sup>ϕ</sup>j*)) *<sup>f</sup>*(*Lim*(*<sup>ϕ</sup>j*)) *<sup>ρ</sup>*31/*<sup>ρ</sup>* <sup>=</sup> 8.74 <sup>1</sup>+(0.200×( <sup>1</sup>−0.252 0.252 )+0.200×( <sup>1</sup>−0.252 0.252 )+...+0.200×( <sup>1</sup>−0.191 0.191 )) <sup>=</sup> 1.609 *Lim*(*A*3) <sup>=</sup> <sup>∑</sup><sup>5</sup> *<sup>j</sup>*=<sup>1</sup> *Lim*(*ϕj*) 1+ 5 ∑ *j*=1 *wj* <sup>1</sup>−*<sup>f</sup>*(*Lim*(*<sup>ϕ</sup>j*)) *<sup>f</sup>*(*Lim*(*<sup>ϕ</sup>j*)) *<sup>ρ</sup>*31/*<sup>ρ</sup>* <sup>=</sup> 13.2 <sup>1</sup>+(0.200×( <sup>1</sup>−0.227 0.227 )+0.200×( <sup>1</sup>−0.227 0.227 )+...+0.200×( <sup>1</sup>−0.189 0.189 )) <sup>=</sup> 2.603

.

In the same way, other values for all alternatives are obtained according to all the criteria, which creates the initial aggregated matrix shown in Table 12.


**Table 12.** Initial aggregated rough matrix.

The summarized values for each criterion, which are necessary for the application of normalization in the next step, are shown in the last row of Table 12. Applying Equation (7), the normalized value for the third alternative according to criterion C11 will be

$$r\_{31} = \left[\frac{1.609}{17.27}; \frac{2.603}{13.05}\right] = [0.09, 0.20].$$

The last row of Table 13 presents the values of the criteria obtained by applying the FUCOM method, which are necessary to create a weighted normalized matrix.



The fourth step is the weighting of normalized rough matrix (Table 14) by multiplying all the values of the normalized matrix with the weights of the criteria by applying Equation (8).

$$d\_{31} = [0.09 \times 0.08; 0.02 \times 0.138] = [0.007, 0.028]$$


**Table 14.** Weighted normalized rough matrix.

The next step is to summarize the values of the alternatives depending on the type of criteria, and two matrices are obtained. The first matrix refers to the sum of the values of alternatives according to the benefit group of criteria, while another one refers to cost criteria. In this research, cost criteria are C11 and C36.

The matrix for alternatives according to benefit criteria is

$$S\_i^+ = \begin{bmatrix} 0.096, & 0.215 \\ 0.122, & 0.274 \\ 0.131, & 0.285 \\ 0.105, & 0.226 \\ 0.104, & 0.240 \end{bmatrix}.$$

An example of the calculation for the third alternative is

$$S\_3^{L+} = \begin{bmatrix} 0.019 + 0.011 + 0.007 + 0.005 + 0.008 + 0.006 + 0.007 + 0.008 + 0.005 + 0.004\\ + 0.003 + 0.006 + 0.004 + 0.005 + 0.005 + 0.010 + 0.004 + 0.010 + 0.003 \end{bmatrix} = [0.131],$$

$$S\_3^{L+} = \begin{bmatrix} 0.043 + 0.023 + 0.017 + 0.011 + 0.019 + 0.012 + 0.014 + 0.016 + 0.008 + 0.007\\ + 0.005 + 0.010 + 0.007 + 0.016 + 0.012 + 0.012 + 0.019 + 0.007 \end{bmatrix} = [0.285].$$

The matrix for alternatives according to cost criteria is

$$S\_i^- = \begin{bmatrix} 0.014, & 0.037\\ 0.015, & 0.044\\ 0.011, & 0.037\\ 0.020, & 0.052\\ 0.018, & 0.053 \end{bmatrix}.$$

An example of calculation is as follows:

$$S\_3^{L-} = [0.007 + 0.004] = [0.011].$$

$$S\_3^{L-} = [0.028 + 0.009] = [0.037].$$

After that, it is necessary to calculate the inverse values of the matrix *S*− *<sup>i</sup>* by applying Equation (11), which is

$$\left(\left(S\_i^{-}\right)^{-1} = \begin{bmatrix} 27.289 & 74.026\\ 22.553 & 64.948\\ 27.353 & 87.059\\ 19.137 & 50.864\\ 19.035 & 54.824 \end{bmatrix} \right)$$

In the next step, it is first necessary to calculate the sum by column for cost criteria applying Equation (12), and the following values are obtained:

$$\left(\overline{S\_i^-}\right) = \sum \left[s\_i^{-L}; s\_i^{-\mathcal{U}}\right] = [0.078, 0.222]\_{\mathcal{V}}$$

and then applying Equation (13) to calculate the sum for the inverse matrix, which will be

$$\left(\overline{S\_i^-}\right)^{-1} = \sum \left[\frac{1}{s\_{ij}^{-\mathrm{U}}}; \frac{1}{s\_{ij}^{-\mathrm{L}}}\right] \\ = [115.367, 331.722].$$

In the next step, it is necessary to determine the relative significance for each alternative. The relative weight *Qi* by alternatives is

$$Q\_{\bar{i}} = \begin{bmatrix} 0.102, & 0.357 \\ 0.127, & 0.399 \\ 0.137, & 0.453 \\ 0.109, & 0.324 \\ 0.108, & 0.346 \end{bmatrix}.$$

The *i*th alternative is calculated using Equation (14). An example of the calculation for the third alternative is

$$Q\_3^L = S\_{3+}^L + \frac{\left(\overline{S\_3^{-L}}\right)}{S\_{3-}^{II} \times \left(\overline{S\_3^{-D}}\right)^{-1}} = 0.131 + \frac{0.078}{0.037 \times 331.722} = 0.137,$$

$$Q\_3^{II} = S\_{3+}^{II} + \frac{\left(\overline{S\_3^{-U}}\right)}{S\_{3-}^{L} \times \left(\overline{S\_3^{-L}}\right)^{-1}} = 0.285 + \frac{0.222}{0.011 \times 115.367} = 0.453.$$

In the last step, the alternatives are ranked from the highest to the lowest value, and the results are as follows: *A*<sup>3</sup> > *A*<sup>2</sup> > *A*<sup>1</sup> > *A*<sup>5</sup> > *A*4.

#### **5. Sensitivity Analysis and Discussion**

The sensitivity analysis has been performed throughout four phases, the first of which involves the creation of nine scenarios where the weights of criteria are modeled. The second phase involves the application of different methods, that is, comparative analysis, while the third phase implies the change of the parameter *ρ* into the values of 1–10. The fourth phase includes the application of Spearman's correlation coefficient for the ranks of alternatives throughout the first two phases.

Figure 2 presents the ranks of alternatives throughout nine scenarios. The first scenario implies that all criteria are equally important, while in the second one, the six most important criteria (C11, C12, C13, C16, C22, C33) are reduced by 4%, and others are increased by 2%. In the third set, the six most important criteria are eliminated, and in the fourth one, the most important criteria are increased by 4%, while the rest are reduced by 2%. The fifth scenario involves the elimination of seven least important criteria (C23, C24, C25, C27, C34, C36, and C37). In the sixth set, the criteria that belong to the economic group are reduced by 4%, while the criteria of the social group are proportionally increased. The values

of environmental criteria remain unchanged. The seventh set implies a reverse situation from the aspect of economic and social criteria in relation to the sixth set. In the eighth scenario, decision-making is based only on economic criteria, and in the ninth scenario, only on environmental criteria.

**Figure 2.** Sensitivity analysis by changing the weight values of criteria.

The ranks of alternatives do not change in the fourth, fifth, sixth, and eighth criteria, which implies that the most important criteria play a very important role in the decision-making process in this research. This is confirmed by the fact that there are significant changes in the rankings in the first and third sets when all the criteria are equal, i.e., when the six most important ones are eliminated. In other scenarios there are no significant changes. It is important to emphasize that the two alternatives that represent the best solution, A3 and A2, do not change ranks in any scenario, which implies that they are insensitive to the changes in the significance of the criteria.

Figure 3 shows the comparison of the proposed model with other approaches developed recently: rough WASPAS [57], rough MABAC [58], rough SAW [59], and rough ARAS [60].

**Figure 3.** Comparison of the results of the developed model with other methods.

Observing the results obtained by other methods, the stability of the two best alternatives do not come into question, since they continue to take the first two positions using all the methods. The highest similarities of ranks obtained with rough COPRAS have the alternatives obtained with rough ARAS, where only the first and fifth alternatives change their positions. Slightly bigger changes in ranks are found with other methods.

Table 15 presents the part of the sensitivity analysis that relates to the change of parameter *ρ*.


**Table 15.** Ranks of alternatives depending on the change of parameter *ρ*.

Changing the parameter *ρ* does not change significantly the initial results obtained. For the parameters *ρ* = 1–6, the same ranks are obtained as with the hybrid FUCOM–rough COPRAS model. The only changes in ranks are for parameters *ρ* = 7, *ρ* = 8, and *ρ* = 10 when the fourth and fifth alternative belongs to the same rank, and when *ρ* = 9, the fourth and fifth alternative change their positions while others remain unchanged. Based on the overall sensitivity analysis with the change of parameter *ρ*, it can be concluded that the model is not sensitive to these changes.

At the end of the sensitivity analysis, the calculated Spearman's correlation coefficient for the first two phases is given (Table 16). For the third phase, calculation is not performed, since it is obvious that there is almost a complete correlation and, as already mentioned, the change of this parameter does not significantly affect the ranking of the alternatives.


**Table 16.** Spearman's correlation coefficient for the first two phases of sensitivity analysis.

Concerning the first phase of the sensitivity analysis in which the weights of the criteria change in sets, it can be seen that the model is sensitive to their changes. The initial set has a full correlation with four sets (4, 5, 6, and 8), while the smallest correlation SCC = 0.600 is with the first and third set, in which the ranks of two alternatives change for a total of three positions. In the second and seventh sets, the two last alternatives change positions between each other, so SCC = 0.900 with the initial set. In the ninth set, there is a change in the rank of three alternatives with SCC = 0.700. The total average value of SCC is 0.856, which represents a high correlation of ranks, regardless of the changes mentioned.

In the second phase, it can be observed that rough COPRAS has the highest correlation with the rough ARAS method of 0.900, while with other methods, rough WASPAS, rough SAW, and rough MABAC, SCC = 0.700. Taking this into account, it is concluded that rough WASPAS, rough SAW, and rough MABAC have a complete correlation, which ultimately implies that the average SCC = 0.920, which is a very high correlation of ranks.

#### **6. Conclusions**

This paper has proposed a new hybrid model that integrates FUCOM with the rough COPRAS method using the rough Dombi aggregator. This is the first time in the literature that this kind of model has been applied, that integrates the positive aspects of FUCOM method, rough set theory, rough Dombi aggregator for group decision-making, and the COPRAS method, which is one of the main contributions of this paper. In addition, the detailed and demonstrated algorithm of the rough COPRAS method also contributes to the overall field of multi-criteria decision-making.

Based on the 21 criteria of sustainability, a total of five suppliers in a construction company were considered, where it was concluded that the third and second suppliers are the best solutions regardless of any change in the model. This has been proven throughout a comprehensive sensitivity analysis in which different scenarios—with a change in the weight of criteria—were formed. The two mentioned alternatives are not sensitive to any changes in the values of the criteria. In addition, neither the change of parameter *ρ*, which is an integral part of the rough Dombi aggregator, affects the rankings of the third and second supplier, which has been confirmed by comparison with other approaches. The best solution in this model is completely insensitive, i.e., stable, while the ranks of other alternatives vary depending on the method of modeling the sensitivity analysis.

The developed model can be useful in other areas of engineering, but also when making real life decisions, since it adequately treats uncertainties by applying the theory of rough sets and subjectivity by applying the FUCOM method. Thus, it is possible to make more accurate and valid decisions that can have a huge impact on a sustainable supply chain. Future research related to this study will address the development and application of a similar model with the FUCOM method and an uncertainty theory, e.g., grey theory.

**Author Contributions:** Each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. Conceptualization, S.S. and B.M.; methodology, Ž.S., S.S. and E.K.Z.; validation, D.K.D., and S.J.; formal analysis, Ž.S. and E.K.Z.; investigation, S.J. and M.M.; writing—original draft preparation, M.M.; writing—review and editing, E.K.Z. and D.K.D.; supervision, B.M.; project administration, B.M.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors acknowledge the support of research projects TR 36017, funded by the Ministry of Science and Technological Development of Serbia.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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