Linguistic intuitionistic fuzzy PROMETHEE method based on similarity measure for the selection of sustainable building materials

Abstract

The selection of sustainable building materials has attracted much attention from society because it is essential for environment, economy, and human health. To meet the need of the building development, the identified criteria for optimal sustainable building materials are determined based on sustainability building standards and previous studies including economic, environmental, social, and technological aspects. To represent the qualitative preferred and non-preferred cognitions of the decision makers, linguistic intuitionistic fuzzy numbers (LIFNs) are utilized to describe the evaluation information, which is a powerful and flexible tool. Then, new accuracy function and score function are defined to rank LIFNs more objectively. In addition, this paper employs a new linguistic intuitionistic fuzzy entropy to calculate the weights of criteria, and models based on the Shapley function with respect to 2-additive measure are constructed to reflect the correlations among elements in a set. Based on these results, a linguistic intuitionistic fuzzy Preference Ranking Organization Method for Enrichment Evaluation and prospect theory based hybrid method is proposed to assess the sustainable building materials. Finally, the effectiveness of the new method is testified by a case study for sustainable indoor flooring material selection, and comparative analysis is made.

Introduction

The construction industry has made a great contribution to society and economy such as providing employment opportunities and promoting economic growth (Plessis 2007). On the other hand, it causes some negative effects such as consuming natural resources and environment pollution (Pulselli et al. 2007; Wang et al. 2005). These aspects are bad for human health (Ries et al. 2006). In order to mitigate the impact of the buildings in the entire life cycle, sustainable construction is put forward as a new building philosophy, which was first proposed by Charles Kibert in 1994. Its main purpose is to solve the ecological and social problems of the construction industry. In 2004, the International Council for Research and Innovation in Building and Construction (CIB) defined sustainable construction as “sustainable production, use, maintenance, demolition and reuse of buildings and constructions or their components” (Drejeris and Kavolynas 2014). The concept of sustainable construction governs three major aspects: environmental protection, social well-being and economic prosperity. The selection of sustainable building materials plays an essential role in the achievement of “sustainable building” (Behzadian et al. 2012), which has been identified as the easiest way for designers to begin incorporating sustainable principles into building projects (Godfaurd et al. 2005). Materials also have an important impact on the environment and public health (Guo and Guo 2011). Currently, there are several sustainable building rating systems such as British Building Research Establishment Environmental Assessment Method (BREEAM), Leadership in Energy and Environmental Design (LEED) and Global Sustainability Assessment System (GSAS) (Awadh 2017). These systems are designed to promote more sustainable building design, construction and operations, which are often adopted to solve the selection of building materials based on credits and points such as minimum recovery, regional materials or made from renewable resources (Tarantini et al. 2011). These rating systems illustrate that materials are crucial for sustainable building study. Generally speaking, “sustainability” stands for a long-term development which integrates four interlinked and inseparable dimensions: economy, ecology, society and technology (Murata 2008). The selection of sustainable building materials is a complex and comprehensive problem that involves many factors such as durability, lifecycle cost, resident support and feasible technology. As stated above, the selection of sustainable building materials in fact is a multiple criteria decision making (MCDM) problem. In this procedure, the types of materials and actual needs should be considered.

There is a need to develop a systematic and holistic sustainable building material selection process for identifying and prioritizing relevant criteria and evaluating alternatives (Akadiri and Olomolaiye 2012). Various MCDM methods for selecting the most suitable alternative based on criteria are proposed. In general, these MCDM methods can be divided into two types: (1) synthetical evaluation methods. For example, Rashid et al. (2017) separately untiled the Analytical Hierarchy Process (AHP) and Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method for ranking sustainable concrete, which assumes that the weights are completely known. Falqi et al. (2019) applied TOPSIS method to order siliceous concrete materials based on sustainable criteria. Reddy et al. (2019) proposed a fuzzy entropy based TOPSIS method for the selection of sustainable building materials. Roy et al. (2019) introduced an interval-valued intuitionistic fuzzy combinative distance assessment (CODAS) method for evaluating sustainable materials in construction projects; (2) integrated methods. For instance, Zavadskas et al. (2017) developed a Step-wise Weight Assessment Ratio Analysis (SWARA) and Multi-objective Optimization by Ratio Analysis Plus Full Multiplicative Form (MULTIMOORA) based hybrid MCDM method for sustainability residential house construction material and element selection, in which SWARA method is applied to calculate the weights of the criteria, and the single-valued neutrosophic MULTIMOORA method is proposed to deal with the indeterminacy of the initial information. Tian et al. (2018) proposed an AHP and grey correlation TOPSIS based hybrid MCDM method for green material selection, where the weights of hierarchical index structure are calculated by AHP method. This method can avoid the TOPSIS method’s subjectivity and irrationality.

It is noticeable that all of the above studies about sustainable building material evaluation use numerical judgments, which cannot deal with fuzzy environments. To address this issue, Zadeh’s fuzzy sets (Zadeh 1965) are good tools. With the development of fuzzy set theory, people realize that quantity fuzzy sets are still insufficient to cope with complex situations. Zadeh (1975) first noted this problem and defined the concept of linguistic variables (LVs). Later, Herrera and Herrera-Viedma (2000) analyzed the characteristics and semantics of linguistic term set (LTS) and presented a linguistic decision-making method. To express the qualitative preferred and non-preferred judgments simultaneously, Chen et al. (2015) proposed the concept of linguistic intuitionistic fuzzy numbers (LIFNs), which uses a LV to denote the membership degree and non-membership degree, respectively. Considering the advantages of LIFNs, several linguistic intuitionistic fuzzy decision-making methods are proposed. For example, Chen et al. (2015) defined some aggregation operators, involving the linguistic intuitionistic fuzzy weighted averaging (LIFWA) operator, linguistic intuitionistic fuzzy ordered weighted averaging (LIFOWA) operator, linguistic intuitionistic fuzzy hybrid weighted averaging (LIFHWA) operator, and their corresponding geometric mean operators. Peng et al. (2018) presented the linguistic intuitionistic fuzzy Frank improved weighted Heronian mean (LIFFIWHM) operator. Tang and Meng (2019) proposed two types of linguistic intuitionistic fuzzy Hamacher aggregation operators including the linguistic intuitionistic fuzzy Hamacher weighted average (LIFHWA) operator and the linguistic intuitionistic fuzzy Hamacher weighted geometric mean (LIFHWGM) operator. To reflect the interactions between elements, Yuan et al. (2019) defined several linguistic intuitionistic fuzzy Shapley aggregation operators. Liu and Liu (2019) applied the linguistic intuitionistic fuzzy partitioned Bonferroni mean (LIFPBM) operator to handle the complex interaction between multiple criteria. Rong et al. (2020) introduced several linguistic intuitionistic Muirhead mean operators, such as the linguistic intuitionistic fuzzy Muirhead mean (LIFMM) operator, the weighted linguistic intuitionistic fuzzy Muirhead mean (WLIFMM) operator and their dual operators, which consider the correlations among criteria and cover other common operators by assigning the different parameter vectors. Besides these aggregation operators based decision-making methods, Li et al. (2017) defined the subtraction and division of LIFNs and proposed an extended Visekriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method based on the entropy measure. Ou et al. (2018) proposed the linguistic intuitionistic fuzzy TOPSIS method. Noticeably, these methods assume that weighting information is completely known. Zhang et al. (2017) researched the weights of the criteria for linguistic intuitionistic fuzzy MCDM using distance measure with the aid of linguistic scale function (LSF) and proposed an extended outranking approach. Meng et al. (2019) introduced LIFNs to preference relations and proposed linguistic intuitionistic fuzzy preference relations (LIFPRs). Then, the authors developed a MCDM method with LIFPRs. Liu et al. (2020) proposed a linguistic intuitionistic fuzzy multi-attribute group decision-making method based on Dempster-Shafer Evidence Theory (DST), which can deal with the situation where both the expert weights and attribute weights are unknown.

As we know, some MCDM methods for selecting sustainable building materials are proposed. However, there are still some drawbacks:

  1. 1.

    It is unreasonable to consider all criteria to be equal importance and/or to regard the weights of criteria to be independent.

  2. 2.

    Most methods for the selection of sustainable building materials are based on the accuracy value, which cannot cope with the fuzzy judgments of DMs.

  3. 3.

    Previous methods for selecting sustainable building materials disregard the risk preference of DMs.

  4. 4.

    All methods for the selection of sustainable building materials cannot denote the qualitative preferred and non-preferred preferences of the DMs.

To address these limitations, this paper proposes a new linguistic intuitionistic fuzzy hybrid method in view of the advantages of Preference Ranking Organization Method for Enrichment Evaluation (PROMETHEE) method and prospect theory (PT). It is notable that the new method considers the attitude of the DMs and cope with the situations where the weighting information is incomplete and interactive. Then, the new method is used to assess the sustainable building materials. To show the merits of the new method, the theoretical and applicable comparison and analysis are made. The main contributions include:

  1. 1.

    The new method employs LIFNs to denote the qualitative preferred and non-preferred judgments, which is more flexible than previous methods for selecting sustainable building materials.

  2. 2.

    To rank LIFNs reasonably, new accuracy function and score function are defined, which consider the attitude of the DMs.

  3. 3.

    To address the issues of previous linguistic intuitionistic fuzzy entropy, a new entropy is defined, by which a linguistic intuitionistic fuzzy similarity measure is derived.

  4. 4.

    To cope with the situations where the weighting information with interactions is incompletely known, models for deriving the fuzzy measure on criteria set are constructed, which are based on the new similarity measure and the Shapley function with respect to 2-additive measure.

  5. 5.

    To describe the risk decision behavior, a linguistic intuitionistic fuzzy PROMETHEE and PT based hybrid method is developed for selecting sustainable building materials.

The rest parts are given as follows: Sect. 2 describes the background of sustainable building materials and builds the evaluating criteria. Section 3 reviews PROMETHEE method, PT, the Shapley function with respect to 2-additive fuzzy measure and LIFNs. Section 4 presents a new score function and a new accuracy function, which improve the discrimination of LIFNs. Meanwhile, a new linguistic intuitionistic fuzzy entropy and a new similarity measure are defined. Section 5 introduces a linguistic intuitionistic fuzzy PROMETHEE and PT based hybrid method for selecting sustainable building materials, which can deal with the cases where the weighting information is incompletely known and interactive. Section 6 uses a case study about the selection of sustainable indoor flooring materials to indicate the application of the new method, and the comparison with other methods is offered. Conclusions are drawn in Sect. 7.

Background of sustainable building materials and evaluation criteria

The construction of urbanization has promoted the rapid development of infrastructures all around the world, in which the construction industry is one of the fastest-growing sectors (Akadiri and Olomolaiye 2012). Buildings are used in residential, official, and commercial activities that play an important role in the health and comfort of humans (Chen et al. 2019). Furthermore, construction industry has created many employment opportunities and made tremendous contributions to the socio-economic development (Zuo and Zhao 2014). However, the development of construction industry is responsible for a considerable proportion of non-renewable resources’ consumption and the pollution of air, soil and water (Mahmoudkelaye et al. 2018). According to United Nations Environment Program (UNEP) and Organisation for Economic Cooperation and Development (OECD), buildings account for 25–40% of total energy consumption, 30–40% of the total solid waste burden, and 30–40% of the carbon dioxide emissions (Huedo et al. 2016). In view of the negative impact on environmental issues and limited resources, a lot of investigations have been conducted to minimize the negative impact on human health and the natural environment, and sustainable construction has attracted broad attention. Sustainable construction aims to satisfy human needs as well as protecting environment, reducing waste of resources, and keeping human comfort and health. The pace of actions towards sustainable application strongly depends on the decisions of stakeholders including owners, managers, designers, companies, etc. The selection of building materials is considered as the easiest way for designers to achieve the aim of sustainable construction throughout the entire life-cycle (Akadiri et al. 2013). The sustainable building materials are eco-friendly, resource-efficient and have little or no harm to human health throughout the life-cycle (Chen et al. 2019). Compared with non-sustainable materials, sustainable building materials consider more environmental and social factors, which makes their selection more complicated.

Usually, the material selection process considers some typical points such as quality, performance, aesthetics and cost. Nowadays, this process has paid more attention to sustainable performance criteria. Many tools are put forward to assess the sustainability of building materials. However, most of these tools focus on environmental sustainability and ignore other aspects, such as economic or social sustainability (Khoshnava et al. 2018). It is necessary to properly consider the assessment criteria and complex relationships between them. In the previous literature, the evaluation criteria for sustainable building materials are constructed through case studies. Abeysundara et al. (2009) developed a quantitative model to select the sustainable building materials, which considers environmental, economic and social assessments of materials from a life cycle perspective. Akadiri et al. (2013) identified six assessment criteria (environmental impact, resource efficiency, waste minimization, life cycle cost, performance capability and social benefit) for the selection of sustainable building materials and applied a fuzzy AHP method to calculate the relative importance of the identified criteria. Considering the mutual effects of evaluation criteria for sustainable building materials, Govindan et al. (2016) utilized a hybrid MCDM method to resolve multiple incompatible and conflicting sustainable building material criteria from three aspects: environment, economy and society. Khoshnava et al. (2018) identified five criteria in the use of green materials for building projects, including resource efficiency, indoor air quality, energy efficiency, water conservation and affordability. The above studies analyze the assessment of sustainable building materials from the perspective of environment, economy and society. However, with the development of technology, people realized that the importance of technology cannot be ignored. Mahmoudkelaye et al. (2018) divided the main criteria into four groups including economic, technical, socio-cultural and environmental factors. Furthermore, Tian et al. (2018) comprehensively summarized technology, environment, economy and society according to previous studies and established the hierarchical index structure for green decoration materials based on interior environmental characteristics.

After reviewing previous studies about the criteria for evaluating sustainable building materials, one can check that they mainly include four first-class evaluation indices such as economy, environment, society and technology. Each of these four first-class evaluation indices further contains several sub-criteria. To show these criteria clearly, we summarize them as shown in Table 1. It should be noted that the evaluation criteria and their importance vary for different types of sustainable building material assessments.

Table 1 The hierarchy of criteria for evaluating sustainable building materials

Basic knowledge

This section includes four parts, the first part introduces PROMETHEE method, the second part reviews PT, the third part recalls fuzzy measure and the Shapley function, and the last part offers some basic concepts about LIFNs.

PROMETHEE method

To improve decision-making quality and reduce evaluation deviations, many MCDM methods are proposed, among of which PROMETHEE method (Brans 1982) is one of the most widely used and powerful decision-making methods. It is a non-compensation and efficient sorting approach that owns some merits such as simplicity, clearness and stability (Brans et al. 1986). It evaluates the alternatives according to the gap between the evaluation values of alternatives and adopts the net flow for ranking, which avoids the compensation and reflects the relative preference of alternatives (Tian et al. 2019). Furthermore, decision makers (DMs) can choose the preference function according to the practical situation. PROMETHEE method was first proposed by Brans (1982) in 1982, which can obtain a partial ranking (PROMETHEE I) or complete ranking (PROMETHEE II) based on the pairwise comparison of alternatives. It is an effective outranking method to solve MCDM problems. The main steps of the classical PROMETHEE method can be summarized as follows:

Step 1:

Define the set of alternatives \(A = \left\{ {a_{1} ,a_{2} ,\ldots,a_{m} } \right\}\) and the set of criteria \(C = \left\{ {c_{1} ,c_{2} ,\ldots,c_{n} } \right\}\), and calculate the preferred value \(P_{k} (a_{i} ,a_{j} )\) of the alternative \(a_{i}\) over \(a_{j}\), where

$$ P_{k} (a_{i} ,a_{j} ) = f_{k} [d_{k} (a_{i} ,a_{j} )] $$
(1)

\(f_{k}\) is a preferred function with the range of [0, 1], \(d_{k} (a_{i} ,a_{j} )\) is the difference between the assessments of the alternatives \(a_{i}\) and \(a_{j}\) for the criterion \(c_{k}\) for all i, j = 1, 2, …, m, and all k = 1, 2, …, n.

Step 2:

Calculate the weighted preferred degree \(\Gamma _{k} (a_{i} ,a_{j} )\) of the alternative \(a_{i}\) over \(a_{j}\) for the criterion \(c_{k}\), expressed as

$$ \Gamma _{k} (a_{i} ,a_{j} ) = w_{k} P_{k} (a_{i} ,a_{j} ) $$
(2)

where \(w_{k}\) is the weight of criterion \(c_{k}\) for all k = 1, 2, …, n.

Step 3:

Calculate the positive net outranking flow \(\phi ^{ + } (a_{i} )\) and the negative net outranking flow \(\phi ^{ - } (a_{i} )\) of the alternative \(a_{i}\), where

$$ \phi ^{ + } (a_{i} ) = \sum\nolimits_{{a_{j} \in A}} {\Gamma _{k} (a_{i} ,a_{j} )} $$
(3)
$$ \phi ^{ - } (a_{i} ) = \sum\nolimits_{{a_{j} \in A}} {\Gamma _{k} (a_{j} ,a_{i} )} $$
(4)

for all i = 1, 2, …, m.

Step 4:

Calculate the net flow of the alternative ai, where

$$ \phi (a_{i} ) = \phi ^{ + } (a_{i} ) - \phi ^{ - } (a_{i} ) $$
(5)

for all i = 1, 2, …, m.

Step 5:

Rank alternatives ai, i = 1, 2, …, m, according to their net flows obtained from Eq. (5).

Prospect theory

In 1979, Kahneman and Tversky (1979) proposed PT to describe human decision-making behavior under risk. PT assumes that individuals are risk aversion for gains and risk pursuit for losses. In PT, the value function plays a key role, which owns three characteristics: (1) the value function can be defined as the deviation from the given reference point; (2) for losses and gains, it is convex and concave, respectively; (3) it is steeper for losses than for gains.

The explicit expression of the value function is

$$ \nu (x) = \left\{ {\begin{array}{*{20}c} {x^{\alpha } \begin{array}{*{20}c} {} & {{\text{ }}} \\ \end{array} {\text{ }}if{\text{ x}} \ge {\text{0}}} \\ { - \lambda ( - x)^{\beta } {\text{ }}if{\text{ x < 0}}} \\ \end{array} } \right. $$
(6)

where α and β such that 0 ≤ α ≤ β ≤ 1 are parameters related to gains and losses, respectively, the parameter λ denotes the degree of loss aversion, which is usually larger than 1. Kahneman and Tversky (1979) experimentally determined the values of α, β and λ, where α = β = 0.88, and λ = 2.25. Furthermore, they suggested that λ belongs to [2, 2.5]. The value function of PT can be described by an S-shaped function as shown in Fig. 1.

Fig. 1
figure1

The value function of PT

Fuzzy measure and the Shapley function

Considering the interactions among elements, fuzzy measure is a good choice, which is defined on the power set and can be seen as an extension of additive measure.

Definition 1

(Grabisch 1997) A fuzzy measure μ on finite set N = {1, 2, …, n} is a set function \(\mu\): P(N) → [0,1] satisfying

  1. (i)

    \(\mu \left( \emptyset \right) = 0,\mu \left( N \right) = 1\),

  2. (ii)

    If A,B ∈ P(N) and \(A \subseteq B\), then \(\mu \left( A \right) \le \mu \left( B \right)\), where P(N) is the power set of N.

To simplify the solving complexity of fuzzy measure, 2-additive measure is one of the most widely used special cases. μ is called a 2-additive measure (Grabisch 1997), if, for any \(S \subseteq N\) with \(s \ge 2\), we have

$$ \mu (S) = \sum\limits_{{\left\{ {i,j} \right\} \subseteq S}} {\mu (i,j) - (s - 2)} \sum\limits_{{i \in S}} {\mu (i)} $$
(7)

where s is the cardinality of S, \(\mu (i)\) and \(\mu (i,j)\) are the fuzzy measures of the coalitions {i} and {i, j} for all i, j = 1, 2, …, n.

With respect to 2-additive measure μ, the Shapley function is a good tool to measure the expect contribution of elements, which is expressed as follows (Meng and Tang 2013):

$$ \Phi _{i} (\mu ,N) = \frac{{3 - n}}{2}\mu (i) + \frac{1}{2}\sum\limits_{{j \in N\backslash i}} {(\mu (i,j) - \mu (j))} $$
(8)

where n is the cardinality of N, and other notations as shown in Eq. (7).

Some concepts about LIFNs

In some practical decision-making problems, LVs are more suitable than quantitative judgments to express the subjective recognitions of the DMs, such as “fair”, “low”, and “high”. To simplify the utilization of LVs, Herrera and Martinez (2000) presented LTSs to denote LVs.

Definition 2

(Herrera and Martinez 2000) A LTS S may be expressed as S = {\(s_{i}\)| i = 0, 1, …, 2t}, where t is a positive integer. Any \(s_{i}\) ∈ S denotes a possible value for a LV, and it should satisfy the following four properties:

  1. (i)

    The set is ordered: \(s_{i} \ge s_{j}\) if \(i \ge j\);

  2. (ii)

    Max operator: \(\max \left( {s_{i} ,s_{j} } \right) = s_{i}\) if \(s_{i} \ge s_{j}\);

  3. (iii)

    Min operator: \(\min \left( {s_{i} ,s_{j} } \right) = s_{i}\) if \(s_{i} \le s_{j}\);

  4. (iv)

    Negation operator: \(neg\left( {s_{i} } \right) = s_{j}\), where j = 2ti.

For example, a LTS S may be defined as: S = {\(s_{0}\): extremely low; \(s_{1}\): very low; \(s_{2}\): low, \(s_{3}\): fair; \(s_{4}\): high; \(s_{5}\): very high; \(s_{6}\): extremely high}. To preserve all of the given information, Xu (2004) extended the discrete LTS S to the continuous LTS \(S_{c} = \left\{ {s_{\alpha } {\text{|}}\alpha \in \left[ {0,2t} \right]} \right\}\).

Definition 3

(Xu 2004) Let S = {\(s_{i}\)| i = 0, 1, …, 2t} be a set of extended continuous linguistic terms, where t is a positive integer. If \(s_{\alpha } \in S\), then \(s_{\alpha }\) is called an original linguistic term. Otherwise, it is called a virtual linguistic term. And the corresponding term index \(\alpha\) can be got by the function I: S → [0, 2t], where \(I(s_{\alpha } ) = \alpha\) for any \(s_{\alpha } \in S\).

Let \(s_{\alpha }\) and \(s_{\beta }\) be any two LVs, then several of their operational laws are defined as follows (Xu 2004):

  1. 1.

    \(s_{\alpha } \oplus s_{\beta } = s_{{\alpha + \beta }}\);

  2. 2.

    \(s_{\alpha }\)\(s_{\beta } = s_{{\alpha - \beta }}\);

  3. 3.

    \(\lambda s_{\alpha } = s_{{\lambda \alpha }}\), \(\lambda \in [0,1]\);

  4. 4.

    \(\lambda (s_{\alpha } \oplus s_{\beta } ) = \lambda s_{\alpha } \oplus \lambda s_{\beta }\), \(\lambda \in [0,1]\);

  5. 5.

    \((\lambda _{1} + \lambda _{2} )s_{\alpha } = \lambda _{1} s_{\alpha } \oplus \lambda _{2} s_{\alpha }\), \(\lambda _{1} ,\lambda _{2} \in [0,1]\).

Although LVs are useful to denote the qualitative judgements, it cannot express the qualitative non-preferred degree. To address this issue, Chen et al. (2015) introduced LIFNs as follows:

Definition 4

(Chen et al. 2015) A LIFN on \(S_{c} = \left\{ {s_{\alpha } {\text{|}}\alpha \in \left[ {0,2t} \right]} \right\}\) is expressed as \(\tilde{\gamma } = (s_{\alpha } ,s_{\beta } )\), where \(s_{\alpha }\) and \(s_{\beta }\) are the qualitative preferred and non-preferred degrees, respectively, and \(s_{\alpha } \oplus s_{\beta } \le s_{{2t}}\).

Considering the order relationship between LIFNs, Chen et al. (2015) introduced the following ranking method.

Definition 5

(Chen et al. 2015) Let \(\tilde{\gamma } = (s_{\alpha } ,s_{\beta } )\) be a LIFN on \(S_{c} = \left\{ {s_{\alpha } {\text{|}}\alpha \in \left[ {0,2t} \right]} \right\}\). Then, the score function is defined as \(S_{C} (\tilde{\gamma }) = \alpha - \beta\) and the accuracy function is defined as \(H_{C} (\tilde{\gamma }) = \alpha + \beta.\)

Let \(\tilde{\gamma }_{1} = (s_{{\alpha _{1} }} ,s_{{\beta _{1} }} )\) and \(\tilde{\gamma }_{2} = (s_{{\alpha _{2} }} ,s_{{\beta _{2} }} )\) be any two LIFNs on \(S_{c} = \left\{ {s_{\alpha } {\text{|}}\alpha \in \left[ {0,2t} \right]} \right\}\). Then, their order relationship is defined as follows:

  1. (i)

    If \(S_{C} (\tilde{\gamma }_{1} ) < S_{C} (\tilde{\gamma }_{2} )\), then \(\tilde{\gamma }_{1} < \tilde{\gamma }_{2}\);

  2. (ii)

    If \(S_{C} (\tilde{\gamma }_{1} ) = S_{C} (\tilde{\gamma }_{2} )\), then \(\left\{ {\begin{array}{*{20}c} {H_{C} (\tilde{\gamma }_{1} ) = H_{C} (\tilde{\gamma }_{2} ) \to \tilde{\gamma }_{1} = \tilde{\gamma }_{2} } \\ {H_{C} (\tilde{\gamma }_{1} ) < H_{C} (\tilde{\gamma }_{2} ) \to \tilde{\gamma }_{1} < \tilde{\gamma }_{2} } \\ \end{array} } \right.\).

According to the above relationship, one can easily derive the following property.

Property 1

Let \(\tilde{\gamma }_{1} = (s_{{\alpha _{1} }} ,s_{{\beta _{1} }} )\) and \(\tilde{\gamma }_{2} = (s_{{\alpha _{2} }} ,s_{{\beta _{2} }} )\) be any two LIFNs on \(S_{c} = \left\{ {s_{\alpha } {\text{|}}\alpha \in \left[ {0,2t} \right]} \right\}\) . Then, \(\tilde{\gamma }_{1} = \tilde{\gamma }_{2}\) if and only if \(s_{{\alpha _{1} }} {\text{ = }}s_{{\alpha _{2} }}\) and \(s_{{\beta _{1} }} {\text{ = }}s_{{\beta _{2} }}\) .

Considering the application of LIFNs in decision-making problems, the ranking method for LIFNs in Definition 5 (Chen et al. 2015) has some deficiencies.

Example 1

Let S = {\(s_{0}\): extremely poor; \(s_{1}\): very poor; \(s_{2}\): poor; \(s_{3}\): slightly poor; \(s_{4}\): fair; \(s_{5}\): slightly good; \(s_{6}\): good; \(s_{7}\): very good; \(s_{8}\): extremely good} be a LTS. Let \(\tilde{\gamma }_{1} = (s_{4} ,s_{4} )\) and \(\tilde{\gamma }_{2} = (s_{1} ,s_{0} )\) be two LIFNs. Then, their accuracy and score values following Definition 5 are

$$ S_{C} (\tilde{\gamma }_{1} ) = 0,\quad S_{C} (\tilde{\gamma }_{2} ) = 1;\quad H_{C} (\tilde{\gamma }_{1} ) = 8,\quad H_{C} (\tilde{\gamma }_{2} ) = 1. $$

Based on Chen et al.’s ranking method (Chen et al. 2015), we get \(\tilde{\gamma }_{1} < \tilde{\gamma }_{2}\). From the perspective of voting, \(\tilde{\gamma }_{1}\) means 4 out of 8 “approval”, 4 out of 8 “opposition”, and no “abstention”; \(\tilde{\gamma }_{2}\) means 1 out of 8 “approval”, no “opposition”, and 7 out of 8 “abstention”. In this case, one can conclude that the LIFN \(\tilde{\gamma }_{1}\) is better than the LIFN \(\tilde{\gamma }_{2}\).

Peng et al. (2018) noted that the ranking results obtained from Chen et al.’s method may be unreasonable in some cases and proposed another method that uses the LSF.

Definition 6

(Peng et al. 2018) Let \(\tilde{\gamma } = (s_{\alpha } ,s_{\beta } )\) be a LIFN on \(S_{c} = \left\{ {s_{\alpha } {\text{|}}\alpha \in \left[ {0,2t} \right]} \right\}\). Furthermore, let \(\tilde{\gamma }^{ * } = (s_{{2t}} ,s_{0} )\) be the positive ideal point, and \(d_{H} (\tilde{\gamma }^{ * } ,\tilde{\gamma })\) be the Hamming distance between \(\tilde{\gamma }^{ * }\) and \(\tilde{\gamma }\). Then, the measurement function R of \(\tilde{\gamma }\) is defined as:

$$ R(\tilde{\gamma }) = 0.5(1 + f^{ * } (s_{{2t - \mu - \nu }} ))d_{H} (\tilde{\gamma }^{ * } ,\tilde{\gamma }) $$
(9)

where \(f^{ * }\): S→[0, 1] is a strictly monotone increasing function for the subscripts of LVs in S, which is called the LSF (Wang et al. 2014). Based on Definition 6, Peng et al. (2018) concluded that the smaller the value of \(R(\tilde{\gamma })\) is, the better the LIFN \(\tilde{\gamma }\) will be.

However, there are still some limitations of Peng et al.’s method (Peng et al. 2018): (1) different ranking results may be obtained for different LSFs; (2) the counterintuitive ranking results may be obtained. To show these issues, let us consider the following example.

Example 2

Let S={\(s_{0}\): extremely poor; \(s_{1}\): very poor; \(s_{2}\): poor; \(s_{3}\): slightly poor; \(s_{4}\): fair; \(s_{5}\): slightly good; \(s_{6}\): good; \(s_{7}\): very good; \(s_{8}\): extremely good} be a LTS, and let \(\tilde{\gamma }_{1} = (s_{{3.2}} ,s_{{1.6}} )\),\(\tilde{\gamma }_{2} = (s_{{3.2}} ,s_{4} )\), \(\tilde{\gamma }_{3} = (s_{4} ,s_{{2.4}} )\) and \(\tilde{\gamma }_{4} = (s_{{2.3}} ,s_{{4.1}} )\) be four LIFNs. Following Definition 6, we have

  1. (i)

    \(R(\tilde{\gamma }_{1} ) = 0.420\),\(R(\tilde{\gamma }_{2} ) = 0.330\),\(R(\tilde{\gamma }_{3} ) = 0.300\),\(R(\tilde{\gamma }_{4} ) = 0.428\) and \(\tilde{\gamma }_{4} < \tilde{\gamma }_{1} < \tilde{\gamma }_{2} < \tilde{\gamma }_{3}\) for

    \(f_{1} ^{ * } (s_{i} ) = \frac{i}{{2\tau }}(i = 0,1, \ldots ,2\tau )\quad {\text{with}}\quad t = 4\)

  2. (ii)

    \(R(\tilde{\gamma }_{1} ) = 0.460\),\(R(\tilde{\gamma }_{2} ) = 0.350\),\(R(\tilde{\gamma }_{3} ) = 0.366\),\(R(\tilde{\gamma }_{4} ) = 0.454\) and \(\tilde{\gamma }_{1} < \tilde{\gamma }_{4} < \tilde{\gamma }_{3} < \tilde{\gamma }_{2}\) for

    $$ f_{2} ^{ * } (s_{i} ) = \left\{ \begin{gathered} \frac{{a^{\tau } - a^{{\tau - i}} }}{{2a^{\tau } - 2}},i = 0,1,\ldots,\tau \hfill \\ \frac{{a^{\tau } + a^{{\tau - i}} - 2}}{{2a^{\tau } - 2}},i = \tau + 1,\tau + 2,\ldots,2\tau \hfill \\ \end{gathered} \right. $$

    with a = 1.37;

  3. (iii)

    \(R(\tilde{\gamma }_{1} ) = 0.407\),\(R(\tilde{\gamma }_{2} ) = 0.330\),\(R(\tilde{\gamma }_{3} ) = 0.283\),\(R(\tilde{\gamma }_{4} ) = 0.678\) and \(\tilde{\gamma }_{4} < \tilde{\gamma }_{1} < \tilde{\gamma }_{2} < \tilde{\gamma }_{3}\) for

    $$ f_{3} ^{ * } (s_{i} ) = \left\{ \begin{gathered} \frac{{\tau ^{\alpha } - (\tau - i)^{\alpha } }}{{2\tau ^{a} }},i = 0,1,\ldots,\tau \hfill \\ \frac{{\tau ^{\beta } {\text{ + }}(i{\text{ - }}\tau )^{\beta } }}{{2\tau ^{\beta } }},i = \tau + 1,\tau + 2,\ldots,2\tau \hfill \\ \end{gathered} \right. $$

    with \(\alpha = \beta = 0.88\).

With respect to the three LSFs in Example 2, we derive two ranking results. Furthermore, all of them show that the LIFN \(\tilde{\gamma }_{2}\) is better than the LIFN \(\tilde{\gamma }_{1}\). However, it is unreasonable because they have the same qualitative preferred degree while the qualitative non-preferred degree of \(\tilde{\gamma }_{2}\) is bigger than that of \(\tilde{\gamma }_{1}\).

New linguistic intuitionistic fuzzy ranking method and similarity measure

This section first presents a new accuracy function and a new score function to build the ranking orders between LIFNs in terms of the preference attitude. Then, a new linguistic intuitionistic fuzzy similarity measure based on entropy measure is proposed, which is used to determine fuzzy measure on the criteria set.

A new ranking method for LIFNs

Based on the above analysis, one can find that neither of them is sufficient to rank LIFNs. The ranking orders obtained from these two methods may be illogical and lead to wrong choices. Considering this issue, following the work of Lin et al. (2018), we next introduce a new method to rank LIFNs, which can address the issues of previous ones.

Definition 7

Let M be the term index function of LIFN \(\tilde{\gamma } = (s_{\alpha } ,s_{\beta } )\) on \(S_{c} = \left\{ {s_{\alpha } {\text{|}}\alpha \in \left[ {0,2t} \right]} \right\}\), where \(M(\tilde{\gamma }) = (\alpha ,\beta )\). The feasible domain of \(M(\tilde{\gamma })\) is defined as

$$ D_{{M(\tilde{\gamma })}} {\text{ = }}\left\{ {\left( {x,y} \right)|\alpha \le x \le \alpha + \pi ,\beta \le y \le \beta + \pi ,x + y \le 2t} \right\} $$
(10)

where \(s_{\pi } = s_{{2t - \alpha - \beta }}\) is the qualitative hesitancy degree, \(D_{{M(\tilde{\gamma })}}\) is shown in Fig. 2.

Fig. 2
figure2

The feasible domain \(D_{{M(\tilde{\gamma })}}\) with respect to \(\tilde{\gamma }\)

Definition 8

Let \(\tilde{\gamma } = (s_{\alpha } ,s_{\beta } )\) be a LIFN on \(S_{c} = \left\{ {s_{\alpha } {\text{|}}\alpha \in \left[ {0,2t} \right]} \right\}\). The preference attitudinal accuracy function of \(\tilde{\gamma }\) is defined as:

$$ H_{\rho } (\tilde{\gamma }){\text{ = }}\frac{1}{{area(D_{{M(\tilde{\gamma })}} )}}\iint_{{D_{{M(\tilde{\gamma })}} }} {(\rho x + (1 - \rho )y)}dxdy $$
(11)

where \(D_{{M(\tilde{\gamma })}}\) is the feasible domain with respect to \(\tilde{\gamma }\) as shown in Eq. (10) and \(area(D_{{M(\tilde{\gamma })}} )\) is the area of the feasible domain \(D_{{M(\tilde{\gamma })}}\) as shown in Fig. 2. The parameter ρ (ρ∈[0,1]) can be regarded as the level of the DM’s preference attitude towards the preferred qualitative judgment. For example, ρ=0.5 means that the DM has a neutral view for the qualitative preferred and non-preferred judgments. \(\iint_{{D_{{M(\tilde{\gamma })}} }} {(\rho x + (1 - \rho )y)}dxdy\) is equal to the volume of the oblique triangular prism P1 as shown in Fig. 3. Figure 3 indicates that \(H_{\rho } (\tilde{\gamma })\) can be regarded as the average height of the oblique triangular prism P1.

Fig. 3
figure3

The graph of the oblique triangular prism \(P_{1}\)

From Definition 7, we get

$$ area(D_{{M(\tilde{\gamma })}} ) = \frac{1}{2}\pi ^{2} = \frac{1}{2}(2t - \alpha - \beta )^{2} $$
(12)

Since

$$ \begin{gathered} \iint_{{D_{{M(\tilde{\gamma })}} }} {(\rho x + (1 - \rho )y)}dxdy{\text{ = }}\int_{\alpha }^{{2t - \beta }} {\left( {\int_{\beta }^{{2t - x}} {\left( {\rho x + (1 - \rho )y} \right)} dy} \right)} dx \hfill \\ = \frac{1}{6}(2t - \alpha - \beta )^{2} (2t - \alpha + 2\beta ) + \frac{\rho }{2}(2t - \alpha - \beta )^{2} (\alpha - \beta ) \hfill \\ \end{gathered} $$
(13)

The preference attitudinal accuracy function can be equivalently expressed as:

$$ \begin{gathered} H_{\rho } (\tilde{\gamma }){\text{ = }}\frac{{2t - \alpha + 2\beta }}{3}{\text{ + }}\rho (\alpha - \beta ) \hfill \\ {\text{ = }}\rho \frac{{2\alpha - \beta {\text{ + }}2t}}{3} + {\text{(1}} - \rho {\text{)}}\frac{{2\beta - \alpha + 2t}}{3} \hfill \\ \end{gathered} $$
(14)

Remark 1

Because it is different to fix the exact value of parameter ρ, just as Lin et al.’ work (Lin et al. 2018), DM’s preference attitudes and the corresponding values of ρ are shown in Table 2.

Table 2 Preference attitudes and the corresponding values of ρ

Similar to the preference attitudinal accuracy function, the preference attitudinal score function is defined as follows:

Definition 9

Let \(\tilde{\gamma } = (s_{\alpha } ,s_{\beta } )\) be a LIFN on \(S_{c} = \left\{ {s_{\alpha } {\text{|}}\alpha \in \left[ {0,2t} \right]} \right\}\). The preference attitudinal score function of \(\tilde{\gamma }\) is expressed as:

$$ S_{\rho } (\tilde{\gamma }){\text{ = }}\frac{1}{{area(D_{{M(\tilde{\gamma })}} )}}\iint_{{D_{{M(\tilde{\gamma })}} }} {(\rho x - (1 - \rho )y)}dxdy $$
(15)

where the notations as shown in Eq. (11).

The graphs of the cubes P2 and P3 are shown in Fig. 4. \(S_{\rho } (\tilde{\gamma })\) is the volume of the cubes P2 and P3. Figure 4 shows that \(S_{\rho } (\tilde{\gamma })\) can be regarded as the average value of the variable z over \(D_{{M(\tilde{\gamma })}}\).

Fig. 4
figure4

The graphs of cubic P2 and P3

Since

$$ \begin{aligned} \iint_{{D_{{M(\tilde{\gamma })}} }} {(\rho x - (1 - \rho )y)}dxdy & = \int_{\alpha }^{{2t - \beta }} {\left( {\int_{\beta }^{{2t - x}} {\left( {\rho x - (1 - \rho )y} \right)} dy} \right)} dx \\ & = \frac{1}{6}(2t - \alpha - \beta )^{2} (\alpha - 2\beta - 2t) + \frac{\rho }{6}(2t - \alpha - \beta )^{2} (\alpha + \beta + 4t) \\ \end{aligned} $$
(16)

The preference attitudinal score function can be equivalently denoted as

$$ \begin{aligned} S_{\rho } (\tilde{\gamma }) & = \frac{{\alpha - 2\beta - 2t}}{3} + \frac{\rho }{3}(\alpha + \beta + 4t) \\ & = \rho \frac{{2\alpha - \beta + 2t}}{3} + (1 - \rho )\frac{{\alpha - 2\beta - 2t}}{3} \\ \end{aligned} $$
(17)

Following the above defined preference attitudinal accuracy and score functions, we offer the following order relationship between LIFNs:

Let \(\tilde{\gamma }_{1} = (s_{{\alpha _{1} }} ,s_{{\beta _{1} }} )\) and \(\tilde{\gamma }_{2} = (s_{{\alpha _{2} }} ,s_{{\beta _{2} }} )\) be any two LIFNs. Then, their order relationship is defined as follows:

  1. (i)

    If \(S_{\rho } (\tilde{\gamma }_{1} ) < S_{\rho } (\tilde{\gamma }_{2} )\), then \(\tilde{\gamma }_{1} < \tilde{\gamma }_{2}\),

  2. (ii)

    If \(S_{\rho } (\tilde{\gamma }_{1} ) = S_{\rho } (\tilde{\gamma }_{2} )\), then \(\left\{ {\begin{array}{*{20}c} {H_{\rho } (\tilde{\gamma }_{1} ) = H_{\rho } (\tilde{\gamma }_{2} ) \to \tilde{\gamma }_{1} = \tilde{\gamma }_{2} } \\ {H_{\rho } (\tilde{\gamma }_{1} ) < H_{\rho } (\tilde{\gamma }_{2} ) \to \tilde{\gamma }_{1} < \tilde{\gamma }_{2} } \\ \end{array} } \right. .\)

Remark 2

When ρ = 0.5, by Eqs. (14) and (17) we get \(H_{{0.5}} {\text{ = (}}\alpha + \beta + 4t{\text{)/6}}\) and \(S_{{\rho = 0.5}} {\text{ = (}}\alpha - \beta {\text{)/2}}\), which is equivalent to the ranking method of Chen et al. (2015).

Example 3

Let S={\(s_{0}\): extremely poor; \(s_{1}\): very poor; \(s_{2}\): poor; \(s_{3}\): slightly poor; \(s_{4}\): fair; \(s_{5}\): slightly good; \(s_{6}\): good; \(s_{7}\): very good; \(s_{8}\): extremely good} be a LTS, and let \(\tilde{\gamma }_{1} = (s_{3} ,s_{{0.5}} )\),\(\tilde{\gamma }_{2} = (s_{4} ,s_{4} )\), \(\tilde{\gamma }_{3} = (s_{4} ,s_{2} )\) and \(\tilde{\gamma }_{4} = (s_{5} ,s_{3} )\) be four LIFNs. The ranking orders acquired from different methods are displayed in Table 3.

Table 3 Ranking orders obtained from different methods

Table 3 shows that the different ranking orders may be obtained from different methods. From the results, the ranking order obtained from Chen et al.’s method (Chen et al. 2015) is the same as that derived from the new method with ρ = 0.5. \(\tilde{\gamma }_{3} > \tilde{\gamma }_{2}\) always holds by the new method which avoids the limitations of Peng et al.’s method (Peng et al. 2018) mentioned in Example 2. From Definition 9, we know that the preference attitudinal score function considers all the possible values in \(D_{{M(\tilde{\gamma })}}\) to indicate the preference attitudes of DMs, which avoids the loss of information. In the new method, ρ = 0.3 and ρ = 0.7 mean the negative and positive information is moderately preferred, respectively. It is worth remarking that the DMs’ different preference attitudes may impact the final ranking orders of the alternatives. For example, Table 3 shows that when ρ = 0.3, \(\tilde{\gamma }_{1}\) is the best choice, and \(\tilde{\gamma }_{4}\) is the best choice when ρ = 0.7.

New linguistic intuitionistic fuzzy entropy and similarity measure

Entropy and similarity measure are useful tools to measure uncertain information. In this subsection, a new entropy and a new similarity measure of LIFNs are presented. To show the different importance of LIFNs for criteria and to reflect their interactions, the Shapley-weighted similarity measure is further defined. Then, models for deriving the optimal fuzzy measures are established to cope with the case where the weighting information is incompletely known and interactive.

Definition 10

Let \(A = \left\{ {\left. {\left\langle {x_{i} ,s_{{\alpha _{A} (x_{i} )}} ,s_{{\beta _{A} (x_{i} )}} } \right\rangle } \right|x_{i} \in X,i = 1,2,\ldots,n} \right\}\) and \(B = \left\{ {\left. {\left\langle {x_{i} ,s_{{\alpha _{B} (x_{i} )}} ,s_{{\beta _{B} (x_{i} )}} } \right\rangle } \right|x_{i} \in X,i = 1,2,\ldots,n} \right\}\) be two linguistic intuitionistic fuzzy sets (LIFSs). Then,

  1. 1.

    \(A \subseteq B\) if and only if \(s_{{\alpha _{A} (x_{i} )}} \le s_{{\alpha _{B} (x_{i} )}}\),\(s_{{\beta _{A} (x_{i} )}} \ge s_{{\beta _{B} (x_{i} )}}\) for each \(x_{i} \in X\);

  2. 2.

    A = B if and only if \(A \subseteq B\) and \(B \subseteq A\);

  3. 3.

    \(A^{C} = \left\{ {\left. {\left\langle {x_{i} ,s_{{\beta _{A} (x_{i} )}} ,s_{{\alpha _{A} (x_{i} )}} } \right\rangle } \right|x_{i} \in X,i = 1,2,\ldots,n} \right\}\).

Similar to the notion of entropy measure of Atanassov’s intuitionistic fuzzy sets (Burillo and Bustince 1996), the linguistic intuitionistic fuzzy entropy is defined as follows.

Definition 11

Let \(A = \left\{ {\left. {\left\langle {x_{i} ,s_{{\alpha _{A} (x_{i} )}} ,s_{{\beta _{A} (x_{i} )}} } \right\rangle } \right|x_{i} \in X,i = 1,2,\ldots,n} \right\}\) a LIFS. The mapping \(E(A)\) is called a linguistic intuitionistic fuzzy entropy, if it satisfies the following conditions:

  1. (E1)

    \(E(A) = s_{0}\) if and only if A is \(x_{i} = (s_{0} ,s_{{2t}} )\) or \(x_{i} = (s_{{2t}} ,s_{0} )\) for each \(x_{i} \in X;\)

  2. (E2)

    \(E(A) = s_{1}\) if and only if \(s_{{\alpha _{A} (x_{i} )}} {\text{ = }}s_{{\beta _{A} (x_{i} )}}\) for each \(x_{i} \in X;\)

  3. (E3)

    \(E(A){\text{ = }}E(A^{C} );\)

  4. (E4)

    \(E(A) \le E(B)\) if

    $$ A \subseteq B\quad {\text{with}}\quad s_{{\alpha _{B} (x_{i} )}} \le s_{{\beta _{B} (x_{i} )}} \quad {\text{for}}\;{\text{each}}\quad x_{i} \in X, $$
    $${\text{or}}\quad A \supseteq B\quad {\text{with}}\quad s_{{\alpha _{B} (x_{i} )}} \ge s_{{\beta _{B} (x_{i} )}} \quad {\text{for}}\;{\text{each}}\quad x_{i} \in X.$$

Inspired by linguistic intuitionistic fuzzy entropy, we give the following definition of linguistic intuitionistic fuzzy similarity measure.

Definition 12

Let \(A = \left\{ {\left. {\left\langle {x_{i} ,s_{{\alpha _{A} (x_{i} )}} ,s_{{\beta _{A} (x_{i} )}} } \right\rangle } \right|x_{i} \in X,i = 1,2, \ldots ,n} \right\}\) and \(B = \left\{ {\left. {\left\langle {x_{i} ,s_{{\alpha _{B} (x_{i} )}} ,s_{{\beta _{B} (x_{i} )}} } \right\rangle } \right|x_{i} \in X,i = 1,2, \ldots ,n} \right\}\) be two LIFSs. The mapping \(S(A,B)\) is called a linguistic intuitionistic fuzzy similarity measure, if it satisfies the following conditions:

  1. (S1)

    \(s_{0} \le S(A,B) \le s_{1}\);

  2. (S2)

    \(S(A,B) = s_{1}\) if and only if \(A{\text{ = }}B\);

  3. (S3)

    \(S(A,B){\text{ = }}S(B,A)\);

  4. (S4)

    If \(A \subseteq B \subseteq C\), then \(S(A,C) \le S(A,B)\) and \(S(A,C) \le S(B,C)\).

Inspired by Zhang and Jiang’s entropy measure for vague sets (Zhang and Jiang 2008), we define the following entropy measure of LIFSs.

Let \(A = \left\{ {\left. {\left\langle {x_{i} ,s_{{\alpha _{A} (x_{i} )}} ,s_{{\beta _{A} (x_{i} )}} } \right\rangle } \right|x_{i} \in X,i = 1,2, \ldots ,n} \right\}\) be a LIFS. Then, its entropy is defined as:

$$ E(A) = s_{{\frac{1}{n}\sum\limits_{{i = 1}}^{n} {\frac{{\min \left\{ {\alpha _{A} (x_{i} ),\beta _{A} (x_{i} )} \right\}}}{{\max \left\{ {\alpha _{A} (x_{i} ),\beta _{A} (x_{i} )} \right\}}}_{{}} } }} $$
(18)

Theorem 1

Let E be a function defined by Eq. (18), then E is an entropy measure of LIFSs.

Proof

To prove that E is an entropy measure of LIFSs, it only needs to show that Eq. (18) satisfies (E1)–(E4) in Definition 11.

(E1) From \(x_{i} = (s_{0} ,s_{{2t}} )\) or \(x_{i} = (s_{{2t}} ,s_{0} )\), we have

$$ s_{{\min \left\{ {\alpha _{A} (x_{i} ),\beta _{A} (x_{i} )} \right\}}} = s_{0} ,\quad s_{{\max \left\{ {\alpha _{A} (x_{i} ),\beta _{A} (x_{i} )} \right\}}} = s_{{2t}} $$

and

$$ E(A) = s_{0} $$

(E2) If \(s_{{\alpha _{A} (x_{i} )}} {\text{ = }}s_{{\beta _{A} (x_{i} )}}\), we get

$$ s_{{\min \left\{ {\alpha _{A} (x_{i} ),\beta _{A} (x_{i} )} \right\}}} = s_{{\max \left\{ {\alpha _{A} (x_{i} ),\beta _{A} (x_{i} )} \right\}}} $$

Therefore, \(E(A) = s_{1}\).

On the other hand, suppose that \(E(A) = s_{1}\). By Eq. (18), we derive

$$ s_{{\min \left\{ {\alpha _{A} (x_{i} ),\beta _{A} (x_{i} )} \right\}}} = s_{{\max \left\{ {\alpha _{A} (x_{i} ),\beta _{A} (x_{i} )} \right\}}} $$

for each \(x_{i} \in X\).

Thus, \(s_{{\alpha _{A} (x_{i} )}} {\text{ = }}s_{{\beta _{A} (x_{i} )}}\).

(E3) From \(A^{C} = \left\{ {\left\langle {x_{i} ,s_{{\beta _{A} (x_{i} )}} ,s_{{\alpha _{A} (x_{i} )}} } \right\rangle |x_{i} \in X,i = 1,2,\ldots,n} \right\}\) and Eq. (18), it easily gets

$$E(A) = E(A^{C} ).$$

(E4) When \(A \subseteq B\) with \(s_{{\alpha _{B} (x_{i} )}} \le s_{{\beta _{B} (x_{i} )}}\) for each \(x_{i} \in X\), we obtain

$$ s_{{\alpha _{A} (x_{i} )}} \le s_{{\alpha _{B} (x_{i} )}} \le s_{{\beta _{B} (x_{i} )}} \le s_{{\beta _{A} (x_{i} )}} $$

Thus,

$$ s_{{\min \left\{ {\alpha _{A} (x_{i} ),\beta _{A} (x_{i} )} \right\}}} \le s_{{\min \left\{ {\alpha _{B} (x_{i} ),\beta _{B} (x_{i} )} \right\}}} , $$
$$ s_{{\max \left\{ {\alpha _{A} (x_{i} ),\beta _{A} (x_{i} )} \right\}}} \ge s_{{\max \left\{ {\alpha _{B} (x_{i} ),\beta _{B} (x_{i} )} \right\}}} . $$

By Eq. (18), we derive \(E(A) \le E(B).\)

Similarly, when \(A \supseteq B\) with \(s_{{\alpha _{B} (x_{i} )}} \ge s_{{\beta _{B} (x_{i} )}}\) for each \(x_{i} \in X\), one can prove that \(E(A) \le E(B)\). ■

Furthermore, we propose the following method to transform the entropy into the similarity measure for LIFSs.

Let \(A = \left\{ {\left. {\left\langle {x_{i} ,s_{{\alpha _{A} (x_{i} )}} ,s_{{\beta _{A} (x_{i} )}} } \right\rangle } \right|x_{i} \in X,i = 1,2, \ldots ,n} \right\}\) and \(B = \left\{ {\left. {\left\langle {x_{i} ,s_{{\alpha _{B} (x_{i} )}} ,s_{{\beta _{B} (x_{i} )}} } \right\rangle } \right|x_{i} \in X,i = 1,2, \ldots ,n} \right\}\) be two LIFSs. Define

$$ s_{{\alpha _{{AB}} (x_{i} )}} = s_{{\frac{{2t + \min \left\{ {{\text{|}}\alpha _{A} (x_{i} ) - \alpha _{B} (x_{i} )|,{\text{|}}\beta _{A} (x_{i} ) - \beta _{B} (x_{i} )|} \right\}}}{{2 \cdot 2t}}}} $$
(19)
$$ s_{{\beta _{{AB}} (x_{i} )}} = s_{{\frac{{2t - \max \left\{ {{\text{|}}\alpha _{A} (x_{i} ) - \alpha _{B} (x_{i} )|,{\text{|}}\beta _{A} (x_{i} ) - \beta _{B} (x_{i} )|} \right\}}}{{2 \cdot 2t}}}} $$
(20)

Let \(g(A,B) = \left\{ {\left\langle {x_{i} ,s_{{\alpha _{{AB}} (x_{i} )}} ,s_{{\beta _{{AB}} (x_{i} )}} } \right\rangle |x_{i} \in X,i = 1,2, \ldots ,n} \right\}\).

It is not difficult to get that \(g(A,B)\) is a LIFS. Now, we prove that \(SI(A,B) = E(g(A,B))\) is a similarity measure of LIFSs.

Theorem 2:

Let E be an entropy measure of LIFSs defined by Eq. (18). Then, the mapping SI, defined by \(SI(A,B) = E(g(A,B))\) for each pair of LIFSs A and B, is a similarity measure of LIFSs.

Proof

To show that SI is a similarity measure of LIFSs, it only needs to prove that \(E(g(A,B))\) satisfies (S1)–(S4) in Definition 12.

(S1) From \(s_{0} \le E(A) \le s_{1}\) for any LIFS A, we have \(s_{0} \le E(g(A,B)) \le s_{1}\).

(S2) From the definition of the entropy measure of LIFNs, we have \(E(g(A,B)){\text{ = }}s_{1}\) if and only if

$$ s_{{\alpha _{{AB}} (x_{i} )}} {\text{ = }}s_{{\beta _{{AB}} (x_{i} )}} $$

for each \(x_{i} \in X\).

By Eqs. (19) and (20), we have \(s_{{|\alpha _{A} (x_{i} ) - \alpha _{B} (x_{i} )|}} {\text{ = }}s_{{{\text{|}}\beta _{A} (x_{i} ) - \beta _{B} (x_{i} )|}} = s_{0}\) for each \(x_{i} \in X\). Thus, \(A = B\).

(S3) From the definition of \(g(A,B)\), it is obvious that \(g(A,B) = g(B,A)\). Hence, \(E(g(A,B)) = E(g(B,A))\).

(S4) Since \(A \subseteq B \subseteq C\), we have \(s_{{\alpha _{A} (x_{i} )}} \le s_{{\alpha _{B} (x_{i} )}} \le s_{{\alpha _{C} (x_{i} )}} ,s_{{\beta _{A} (x_{i} )}} \ge s_{{\beta _{B} (x_{i} )}} \ge s_{{\beta _{C} (x_{i} )}}\). Therefore,

\(s_{{|\alpha _{A} (x_{i} ) - \alpha _{C} (x_{i} )|}} \ge s_{{|\alpha _{A} (x_{i} ) - \alpha _{B} (x_{i} )|}} ,s_{{{\text{|}}\beta _{A} (x_{i} ) - \beta _{C} (x_{i} )|}} \ge s_{{{\text{|}}\beta _{A} (x_{i} ) - \beta _{B} (x_{i} )|}}\).

It follows that

$$ s_{{\min \left\{ {{\text{|}}\alpha _{A} (x_{i} ) - \alpha _{C} (x_{i} )|,{\text{|}}\beta _{A} (x_{i} ) - \beta _{C} (x_{i} )|} \right\}}} \ge s_{{\min \left\{ {{\text{|}}\alpha _{A} (x_{i} ) - \alpha _{B} (x_{i} )|,{\text{|}}\beta _{A} (x_{i} ) - \beta _{B} (x_{i} )|} \right\}}} , $$

and

$$ s_{{\max \left\{ {{\text{|}}\alpha _{A} (x_{i} ) - \alpha _{C} (x_{i} )|,{\text{|}}\beta _{A} (x_{i} ) - \beta _{C} (x_{i} )|} \right\}}} \ge s_{{\max \left\{ {{\text{|}}\alpha _{A} (x_{i} ) - \alpha _{B} (x_{i} )|,{\text{|}}\beta _{A} (x_{i} ) - \beta _{B} (x_{i} )|} \right\}}} , $$

we get

$$ g(A,B) \subseteq g(A,C). $$

Therefore, \(E(g(A,C)) \ge E(g(A,B))\)

Corollary 1:

Let E be an entropy measure of LIFSs defined by Eq. (18). Then, the similarity mapping SI, given in Theorem 2, i.e., \(SI(A,B) = E(g(A,B))\) for each pair of LIFSs A and B, can be defined as

$$ SI(A,B) = E(g(A,B)) = s_{{\frac{1}{n}\sum\limits_{{i = 1}}^{n} {\frac{{2t - \max \left\{ {{\text{|}}\alpha _{A} (x_{i} ) - \alpha _{B} (x_{i} )|,{\text{|}}\beta _{A} (x_{i} ) - \beta _{B} (x_{i} )|} \right\}}}{{2t + \min \left\{ {{\text{|}}\alpha _{A} (x_{i} ) - \alpha _{B} (x_{i} )|,{\text{|}}\beta _{A} (x_{i} ) - \beta _{B} (x_{i} )|} \right\}}}} }} $$
(21)

Moreover, if there are interactions between elements in X, then we define the following 2-additive Shapley-weighted similarity measure of LIFSs:

$$ SI^{ * } (A,B){\text{ = }}s_{{\frac{1}{n}\sum\limits_{{i = 1}}^{n} {\Phi _{{x_{i} }} (\mu ,X)\frac{{2t - \max \left\{ {{\text{|}}\alpha _{A} (x_{i} ) - \alpha _{B} (x_{i} )|,{\text{|}}\beta _{A} (x_{i} ) - \beta _{B} (x_{i} )|} \right\}}}{{2t + \min \left\{ {{\text{|}}\alpha _{A} (x_{i} ) - \alpha _{B} (x_{i} )|,{\text{|}}\beta _{A} (x_{i} ) - \beta _{B} (x_{i} )|} \right\}}}} }} $$
(22)

where \(\Phi _{{x_{i} }} (\mu ,X)\) is the Shapley value of the element xi with respect to the 2-additive measure μ.

If there is no interaction among elements in X, we get its corresponding weighted similarity measure.

Considering the correlations among criteria, models for deriving the optimal fuzzy measures are established which are based on the new similarity measure and the Shapley function with respect to 2-additive measure. Define the set of alternatives \(A = \left\{ {a_{1} ,a_{2} ,\ldots,a_{m} } \right\}\) and the set of criteria \(C = \left\{ {c_{1} ,c_{2} ,\ldots,c_{n} } \right\}\), and form the linguistic intuitionistic fuzzy matrix (LIFM) \(\tilde{R} = (\tilde{\gamma }_{{ij}} )_{{m \times n}}\). Transform the LIFM \(R = (\tilde{\gamma }_{{ij}} )_{{m \times n}}\) into the normalized LIFM \(R^{\prime} = (\tilde{\gamma }^{\prime}_{{ij}} )_{{m \times n}}\).

where

$$ \tilde{\gamma ^{\prime}}_{{ij}} {\text{ = }}\left\{ {\begin{array}{*{20}c} {\tilde{\gamma }_{{ij}} = (s_{\alpha } ,s_{\beta } )} & {{\text{ for benefit c}}_{j} } \\ {\tilde{\gamma }_{{ij}}^{c} = (s_{\beta } ,s_{\alpha } )} & {{\text{for cost c}}_{j} } \\ \end{array} } \right. $$
(23)

According to the similarity theory and the Shapley-weighted similarity measure, one can see that the smaller the similarity, the bigger the difference between the assessments of the alternatives. Therefore, it can provide DMs with more useful information. Such criteria should be given a bigger weight. Otherwise, the criteria will be considered to be relative unimportance. When the 2-aaditive measure on criteria set C is incompletely known, we establish the following model to ascertain it.

$$ \begin{gathered} \min \sum\limits_{{j = 1}}^{n} {\sum\limits_{{l = 1}}^{n} {I(SI(R^{\prime}_{j} ,R^{\prime}_{l} ))\Phi _{{c_{j} }} (\mu ,C)} } \hfill \\ s.t.\left\{ \begin{gathered} \sum\limits_{{c_{l} \subseteq S\backslash c_{j} }} {(\mu (c_{j} ,c_{l} ) - \mu (c_{l} ))} \ge (|S| - 2)\mu (c_{j} ),\forall S \subseteq C,\forall c_{j} \in S,|S| \ge 2, \hfill \\ \sum\limits_{{\{ c_{j} ,c_{l} \} \subseteq C}} {\mu (c_{j} ,c_{l} ) - (|C| - 2)\sum\limits_{{c_{j} \in C}} {\mu (c_{j} )} } {\text{ = 1,}} \hfill \\ \mu (c_{j} ) \in W_{j} ,\mu (c_{j} ) \ge 0,j = 1,\ldots,n \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered} $$
(24)

Put Eq. (8) into the objective function of model (24), we derive the following model:

$$ \begin{gathered} \min \frac{{3{\text{ - }}n}}{2}\sum\limits_{{j = 1}}^{n} {\sum\limits_{{l = 1}}^{n} {I(SI(R^{\prime}_{j} ,R^{\prime}_{l} )\mu (c_{j} )} } ) + \frac{1}{2}\sum\limits_{{j = 1}}^{n} {\sum\limits_{{l = 1}}^{n} {I(SI(R^{\prime}_{j} ,R^{\prime}_{l} )(\mu (c_{j} ,c_{l} ) - \mu (c_{l} ))} } ) \hfill \\ s.t.\left\{ \begin{gathered} \sum\limits_{{c_{l} \subseteq S\backslash c_{j} }} {(\mu (c_{j} ,c_{l} ) - \mu (c_{l} ))} \ge (|S| - 2)\mu (c_{j} ),\forall S \subseteq C,\forall c_{j} \in S,|S| \ge 2, \hfill \\ \sum\limits_{{\{ c_{j} ,c_{l} \} \subseteq C}} {\mu (c_{j} ,c_{l} ) - (|C| - 2)\sum\limits_{{c_{j} \in C}} {\mu (c_{j} )} } {\text{ = 1,}} \hfill \\ \mu (c_{j} ) \in W_{j} ,\mu (c_{j} ) \ge 0,j = 1,\ldots,n \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered} $$
(25)

where I is the function defined as I: S → [0, 2t], namely, I() = α for any  ∈ [0, 2t], \(\Phi _{{c_{j} }} (\mu ,C)\) is the Shapley value of the criterion cj with respect to the 2-additive fuzzy measure μ, \(R^{\prime}_{j}\) is the jth column of the normalized LIFM \(R^{\prime} = (\tilde{\gamma }^{\prime}_{{ij}} )_{{m \times n}}\), and Wj is the known weighting information of the criterion cj for all j = 1, 2, …, n.

A linguistic intuitionistic fuzzy PROMETHEE method

According to the above analysis, this section offers a MCDM method with linguistic intuitionistic fuzzy information, which can deal with the cases where the weighting information with interactions is incompletely known. The detail decision steps are summarized as follows:

Step 1:

Let \(R^{\prime} = (\tilde{\gamma }^{\prime}_{{ij}} )_{{m \times n}}\) be the normalized LIFM of \(\tilde{R} = (\tilde{\gamma }_{{ij}} )_{{m \times n}}\), where \(\tilde{\gamma }^{\prime}_{{ij}}\) as shown in Eq. (23) for all I = 1, 2, …, m and all j = 1, 2, …, n.

Step 2:

Model (25) is adopted to get the optimal 2-additive measure on the sub-criteria set Ck, where k = 1, 2, …, p, and p is the number of the first class-criteria. Then, Eq. (8) is used to calculate the Shapley value \(\Phi _{{c_{j}^{k} }} (\mu ,C^{k} )\) of the sub-criterion \(c_{j}^{k}\), where j = 1, 2, …, nk, and nk is the number of sub-criteria in Ck.

Step 3:

Determine the weighted preferred degree \(\Gamma ^{k} (a_{i} ,a_{l} )\) of the alternative \(a_{i}\) over \(a_{l}\) for the criterion Ck, which is determined by the preferred value \(P_{j}^{k} (a_{i} ,a_{l} )\):

$$ \Gamma ^{k} (a_{i} ,a_{l} ) = \sum\limits_{{j = 1}}^{{n_{k} }} {\Phi _{{c_{j}^{k} }} (\mu ,C^{k} )P_{j}^{k} (a_{i} ,a_{l} )} $$
(26)

The linear preference function is the most appropriate of the six generalized types, i.e., the linear shape with indifference criterion (Liao and Xu 2014), where

$$ P_{j}^{k} (a_{i} ,a_{l} ) = \left\{ {\begin{array}{*{20}c} 0 & {d_{j}^{k} (a_{i} ,a_{l} ) \le q_{j}^{k} } \\ {\frac{{d_{j}^{k} (a_{i} ,a_{l} ) - q_{j}^{k} }}{{p_{j}^{k} - q_{j}^{k} }}} & {q_{j}^{k} \le d_{j}^{k} (a_{i} ,a_{l} ) \le p_{j}^{k} } \\ 1 & {d_{j}^{k} (a_{i} ,a_{l} ) > p_{j}^{k} } \\ \end{array} } \right. $$
(27)

\(d_{j}^{k} (a_{i} ,a_{l} ){\text{ = }}S_{j}^{k} (a_{i} ) - S_{j}^{k} (a_{l} )\), \(S_{j}^{k} (a_{i} )\) is the new score function of the alternative \(a_{i}\) for the criterion \(c_{j}^{k}\). For each criterion, the parameters \(q_{j}^{k}\) (the value of the indifference threshold of the criterion \(c_{j}^{k}\), namely, the biggest negligible difference) and \(p_{j}^{k}\)(the value of the strict preference threshold of the criterion \(c_{j}^{k}\), that is, the smallest difference to generate a full preference) are defined by the DMs based on the actual situation.

According to Eq. (27), the graphical representation of this preference function is provided in Fig.5.

Fig. 5
figure5

Criterion with linear preference and indifference area

Step 4:

To reflect the risk-averse psychology of DMs, we integrate the PT into the weighted preferred degree \(\Gamma ^{k} (a_{i} ,a_{l} )\). The positive, negative and net flows of each alternative with respect to each first-class criterion are defined as follows:

$$ \phi ^{ + } (a_{i} ) = \sum\nolimits_{{a_{l} \in A}} {\left[ {\Gamma ^{k} (a_{i} ,a_{l} )} \right]} ^{\alpha } $$
(28)
$$ \phi ^{ - } (a_{i} ) = \sum\nolimits_{{a_{l} \in A}} {\lambda \left[ {\Gamma ^{k} (a_{l} ,a_{i} )} \right]} ^{\beta } $$
(29)
$$ \phi (a_{i} ) = \phi ^{ + } (a_{i} ) - \phi ^{ - } (a_{i} ) $$
(30)

where α and β are parameters related to the gain and loss, respectively, and the parameter λ is the degree of loss aversion.

Step 5:

Since the net flows of alternatives the bigger the better, we establish the following model for the optimal fuzzy measure on first-class criteria set C={C1, C2, …, Cp}.

$$ \begin{gathered} \max \sum\limits_{{i = 1}}^{m} {\sum\limits_{{k = 1}}^{p} {\phi _{{ik}} \Phi _{{C^{k} }} (\mu ,C)} } \hfill \\ s.t.\left\{ \begin{gathered} \sum\limits_{{c_{l} \subseteq S\backslash c^{k} }} {(\mu (C^{k} ,C^{v} ) - \mu (C^{v} ))} \ge (|S| - 2)\mu (C^{k} ),\forall S \subseteq C,\forall C^{k} \in S,|S| \ge 2, \hfill \\ \sum\limits_{{\{ c^{k} ,c^{v} \} \subseteq C}} {\mu (C^{k} ,C^{v} ) - (|C| - 2)\sum\limits_{{c^{k} \in C}} {\mu (C^{k} )} } {\text{ = 1,}} \hfill \\ \mu (C^{k} ) \in W^{k} ,\mu (C^{k} ) \ge 0,k = 1,\ldots,p \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered} $$
(31)

where \(\phi _{{ik}}\) is the net flow of the alternative \(a_{i}\) for the first-class criterion Ck. Then, we use Eq. (8) to calculate the Shapley value \(\Phi _{{C^{k} }} (\mu ,C)\) of each first-class criterion Ck, where k = 1, 2, …, p.

Step 6:

Again apply Step 4 to calculate the net flow of each alternative, where the weighted preferred degree is defined as follows:

$$ \Gamma (a_{i} ,a_{l} ) = \sum\limits_{{k = 1}}^{p} {\Phi _{{C^{k} }} (\mu ,C)\Gamma ^{k} (a_{i} ,a_{l} )} $$
(32)

Remark 3:

To cope with the situations where the weighting information with interactive characteristics is incompletely known, models (25) and (31) for deriving the fuzzy measure on criteria set are constructed, which are based on the Shapley function with respect to 2-additive measure. When there are no interactions between criteria, we get models for the optimal additive weight vectors. To calculate the differences between alternatives, new accuracy and score functions are defined. Furthermore, to reflect risk preference of DMs, PT is included to calculate the ranking results.

In this section, we provide a linguistic intuitionistic fuzzy PROMETHEE and PT based hybrid method for ranking and evaluating alternatives. Meanwhile, the new similarity measure and the Shapley function with respect to 2-additive measure are used to acquire the weight vector when it is incompletely known. To show the procedure of the linguistic intuitionistic fuzzy PROMETHEE method, please see Fig. 6.

Fig. 6
figure6

The process of the new method

Case study

In this section, we first offer an example to select the best sustainable indoor flooring materials to illustrate the efficiency of the new method. Then, we compare it with several previous methods about decision making with LIFNs.

A case study

In the face of energy shortage and environmental degradation, the government pays more attention to sustainable development in the long-term plan. The choice of sustainable building materials is an important aspect of sustainable development, which is the easiest way to achieve the sustainable development of construction projects. Various types of sustainable building materials have been extensively studied. Floor is an important component of construction. It not only provides basic functions including sheltering and avoiding temperature fluctuations but also meets the requirements of aesthetics and comfort. Therefore, the selection of sustainable indoor flooring materials is a complex MCDM problem, which considers various factors including economy, ecology, society and technology. There are four types of sustainable indoor flooring materials, which are evaluated as alternatives including terrazzo flooring (\(a_{1}\)), solid hardwood flooring (\(a_{2}\)), luxury vinyl planks (\(a_{3}\)) and ceramic tiles (\(a_{4}\)) (Chen et al. 2019). Figure 7 shows the hierarchical structure of evaluation criteria, and their explanations are offered in Table 1.

Fig. 7
figure7

The hierarchical structure of sustainable building materials

Based on the previous research (Akadiri et al. 2013; Khoshnava et al. 2018; Mahmoudkelaye et al. 2018) and the experience of DMs, the given weighting information is shown in Table 4. Considering the fact that this problem is too complex and too ill-defined for DMs in a quantitative form, the terms like ‘‘good”, ‘‘fair”, and ‘‘poor” are more adequate than numerical values. Therefore, we apply LIFNs to denote the preferred and non-preferred judgments of DMs simultaneously. Let \(S\) = {\(s_{0}\): extremely poor; \(s_{1}\): very poor; \(s_{2}\): poor; \(s_{3}\): slightly poor; \(s_{4}\): fair; \(s_{5}\): slightly good; \(s_{6}\): good; \(s_{7}\): very good; \(s_{8}\): extremely good} be the predefined LTS. The LIFMs offered by DMs are shown in Tables 5, 6, 7, 8. Let us take the DMs’ evaluation for the alternative \(a_{1}\) with respect to the criterion \(c_{1}^{1}\) as an example. The DMs believe that the preferred degree of the alternative \(a_{1}\) for the criterion \(c_{1}^{1}\) is “\(s_{5}\): slightly good”, and the non-preferred degree is “\(s_{3}\): slightly poor”.

Table 4 The known interval weighting information
Table 5 The LIFM \(R_{{(1)}}\) with respect to the criterion C1
Table 6 The LIFM \(R_{{(2)}}\) with respect to the criterion C2
Table 7 The LIFM \(R_{{(3)}}\) with respect to the criterion C3
Table 8 The LIFM \(R_{{(4)}}\) with respect to the criterion C4

To derive the ranking of these four types of sustainable indoor flooring materials based on the new method, the following procedure is required:

Step 1:

Because the criterion C1 is cost, and the criteria C2, C3 and C4 are benefit, we need to transform the LIFM \(R_{{(1)}}\) into the normalized LIFM \(R^{\prime}_{{(1)}}\) by Eq. (23), which is shown in Table 9.

Table 9 The normalized LIFM \(R^{\prime}_{{(1)}}\) with respect to the criterion C1
Step 2:

According to normalized LIFMs and model (25), the following linear programming model is constructed for obtaining the optimal 2-additve measure for the economic sub-criteria set C1.

$$ \begin{gathered} \min - 3.16\mu \left( {c_{1}^{1} } \right) - 2.95\mu \left( {c_{2}^{1} } \right) - 3.19\mu \left( {c_{3}^{1} } \right) + 2.92\mu \left( {c_{1}^{1} ,c_{2}^{1} } \right) + 3.40\mu \left( {c_{1}^{1} ,c_{3}^{1} } \right) + 2.98\mu \left( {c_{2}^{1} ,c_{3}^{1} } \right) \hfill \\ \begin{array}{*{20}c} {} & {} & {s.t.\left\{ \begin{gathered} \mu \left( {c_{1}^{1} } \right) + \mu \left( {c_{2}^{1} } \right) - \mu \left( {c_{1}^{1} ,c_{2}^{1} } \right) \le 0 \hfill \\ \mu \left( {c_{1}^{1} } \right) + \mu \left( {c_{3}^{1} } \right) - \mu \left( {c_{1}^{1} ,c_{3}^{1} } \right) \le 0 \hfill \\ \mu \left( {c_{2}^{1} } \right) + \mu \left( {c_{3}^{1} } \right) - \mu \left( {c_{2}^{1} ,c_{3}^{1} } \right) \le 0 \hfill \\ \mu \left( {c_{1}^{1} ,c_{2}^{1} } \right) + \mu \left( {c_{1}^{1} ,c_{3}^{1} } \right) + \mu \left( {c_{2}^{1} ,c_{3}^{1} } \right) - \mu \left( {c_{1}^{1} } \right) - \mu \left( {c_{2}^{1} } \right) - \mu \left( {c_{3}^{1} } \right) = 1 \hfill \\ \mu \left( {c_{1}^{1} } \right) \in [0.4,0.5],\mu \left( {c_{2}^{1} } \right) \in [0.35,0.4],\mu \left( {c_{3}^{1} } \right) \in [0.1,0.25] \hfill \\ \end{gathered} \right.} \\ \end{array} \hfill \\ \end{gathered} $$

Solving the above model using MATLAB, we obtain

$$ \mu (c_{1}^{1} ) = 0.4,\;\mu (c_{2}^{1} ) = 0.35,\;\mu (c_{3}^{1} ) = 0.1,\;\mu (c_{1}^{1} ,c_{2}^{1} ) = 0.9,\;\mu (c_{1}^{1} ,c_{3}^{1} ) = 0.5,\;\mu (c_{2}^{1} ,c_{3}^{1} ) = 0.45. $$

By Eq. (8), we obtain the economic sub-criteria’ Shapley values as follows:

$$ \Phi _{{c_{1}^{1} }} (\mu ,C^{1} ) = 0.475,\quad \Phi _{{c_{2}^{1} }} (\mu ,C^{1} ) = 0.425,\quad {\text{ }}\Phi _{{c_{3}^{1} }} (\mu ,C^{1} ) = 0.1. $$

Similarly, we obtain the Shapley values of the environmental, social and technological sub-criteria as follows:

$$ \begin{gathered} \Phi _{{c_{1}^{2} }} (\mu ,C^{2} ) = 0.25,\quad \Phi _{{c_{2}^{2} }} (\mu ,C^{2} ) = 0.4,\quad {\text{ }}\Phi _{{c_{3}^{2} }} (\mu ,C^{2} ) = 0.15,\quad {\text{ }}\Phi _{{c_{4}^{2} }} (\mu ,C^{2} ) = 0.1,\quad {\text{ }}\Phi _{{c_{5}^{2} }} (\mu ,C^{2} ) = 0.1, \hfill \\ \Phi _{{c_{1}^{3} }} (\mu ,C^{3} ) = 0.22,\quad \Phi _{{c_{2}^{3} }} (\mu ,C^{3} ) = 0.28,\quad {\text{ }}\Phi _{{c_{3}^{3} }} (\mu ,C^{3} ) = 0.17,\quad {\text{ }}\Phi _{{c_{4}^{3} }} (\mu ,C^{3} ) = 0.33 \hfill \\ \Phi _{{c_{1}^{4} }} (\mu ,C^{4} ) = 0.1,\quad \Phi _{{c_{2}^{4} }} (\mu ,C^{4} ) = 0.15,\quad \Phi _{{c_{3}^{4} }} (\mu ,C^{4} ) = 0.2,\quad \Phi _{{c_{4}^{4} }} (\mu ,C^{4} ) = 0.3,\quad \Phi _{{c_{5}^{4} }} (\mu ,C^{4} ) = 0.25. \hfill \\ \end{gathered} $$
Step 3:

Calculate the new score function \(S_{j}^{k} (a_{i} )\) of the alternative ai for the criterion \(c_{j}^{k}\). The linear preference function is selected for each criterion. The indifference threshold is defined as \(q_{j}^{k} = 0.05 \times [\max .S_{j}^{k} - \min .S_{j}^{k} ]\), and the preference threshold is defined as \(p_{j} = 0.6 \times [\max .S_{j} - \min .S_{j} ]\) (Gervasio and Silva 2012), where \(\max .S_{j}^{k}\) and \(\min .S_{j}^{k}\) are the maximum and minimum score functions for the criterion \(c_{j}^{k}\). In addition, the score function is defined as ρ = 0.5.

Step 4:

Consider the PT with the parameters α = β = 0.88, and λ = 2.25, which are derived from the empirical data in the reference (Kahneman and Tversky 1979). Calculate the positive, negative and net flows of each alternative, respectively.

Step 5:

Based on the net flow, we construct the following programming model to get the optimal fuzzy measure on first-class criteria set C.

$$ \begin{gathered} \max 39.60\mu \left( {C^{1} } \right) + 23.41\mu \left( {C^{2} } \right) + 50.06\mu \left( {C^{3} } \right) + 39.26\mu \left( {C^{4} } \right) - 12.23\mu \left( {C^{1} ,C^{2} } \right) - 38.88\mu \left( {C^{1} ,C^{3} } \right) \hfill \\ \begin{array}{*{20}c} {} & {} & {} \\ \end{array} - 28.08\mu \left( {C^{1} ,C^{4} } \right) - 22.69\mu \left( {C^{2} ,C^{3} } \right) - 11.90\mu \left( {C^{2} ,C^{4} } \right) - 38.55\mu \left( {C^{3} ,C^{4} } \right) \hfill \\ \begin{array}{*{20}c} {} & {} & {s.t.\left\{ \begin{gathered} \mu \left( {C^{1} } \right) + \mu \left( {C^{2} } \right) - \mu \left( {C^{1} ,C^{2} } \right) \le 0,\mu \left( {C^{1} } \right) + \mu \left( {C^{3} } \right) - \mu \left( {C^{1} ,C^{3} } \right) \le 0 \hfill \\ \mu \left( {C^{1} } \right) + \mu \left( {C^{4} } \right) - \mu \left( {C^{1} ,C^{4} } \right) \le 0,\mu \left( {C^{2} } \right) + \mu \left( {C^{3} } \right) - \mu \left( {C^{2} ,C^{3} } \right) \le 0 \hfill \\ \mu \left( {C^{2} } \right) + \mu \left( {C^{4} } \right) - \mu \left( {C^{2} ,C^{4} } \right) \le 0,\mu \left( {C^{3} } \right) + \mu \left( {C^{4} } \right) - \mu \left( {C^{3} ,C^{4} } \right) \le 0 \hfill \\ \mu \left( {C^{1} } \right) + \mu \left( {C^{2} } \right) + \mu \left( {C^{3} } \right) - \mu \left( {C^{1} ,C^{2} } \right) - \mu \left( {C^{1} ,C^{3} } \right) \le 0 \hfill \\ \mu \left( {C^{1} } \right) + \mu \left( {C^{2} } \right) + \mu \left( {C^{3} } \right) - \mu \left( {C^{1} ,C^{2} } \right) - \mu \left( {C^{2} ,C^{3} } \right) \le 0 \hfill \\ \mu \left( {C^{1} } \right) + \mu \left( {C^{2} } \right) + \mu \left( {C^{3} } \right) - \mu \left( {C^{1} ,C^{3} } \right) - \mu \left( {C^{2} ,C^{3} } \right) \le 0 \hfill \\ \mu \left( {C^{1} } \right) + \mu \left( {C^{2} } \right) + \mu \left( {C^{4} } \right) - \mu \left( {C^{1} ,C^{2} } \right) - \mu \left( {C^{1} ,C^{4} } \right) \le 0 \hfill \\ \mu \left( {C^{1} } \right) + \mu \left( {C^{2} } \right) + \mu \left( {C^{4} } \right) - \mu \left( {C^{1} ,C^{2} } \right) - \mu \left( {C^{2} ,C^{4} } \right) \le 0 \hfill \\ \mu \left( {C^{1} } \right) + \mu \left( {C^{2} } \right) + \mu \left( {C^{4} } \right) - \mu \left( {C^{1} ,C^{4} } \right) - \mu \left( {C^{2} ,C^{4} } \right) \le 0 \hfill \\ \mu \left( {C^{1} } \right) + \mu \left( {C^{3} } \right) + \mu \left( {C^{4} } \right) - \mu \left( {C^{1} ,C^{3} } \right) - \mu \left( {C^{1} ,C^{4} } \right) \le 0 \hfill \\ \mu \left( {C^{1} } \right) + \mu \left( {C^{3} } \right) + \mu \left( {C^{4} } \right) - \mu \left( {C^{1} ,C^{3} } \right) - \mu \left( {C^{2} ,C^{4} } \right) \le 0 \hfill \\ \mu \left( {C^{1} } \right) + \mu \left( {C^{3} } \right) + \mu \left( {C^{4} } \right) - \mu \left( {C^{1} ,C^{4} } \right) - \mu \left( {C^{3} ,C^{4} } \right) \le 0 \hfill \\ \mu \left( {C^{2} } \right) + \mu \left( {C^{3} } \right) + \mu \left( {C^{4} } \right) - \mu \left( {C^{2} ,C^{3} } \right) - \mu \left( {C^{2} ,C^{4} } \right) \le 0 \hfill \\ \mu \left( {C^{2} } \right) + \mu \left( {C^{3} } \right) + \mu \left( {C^{4} } \right) - \mu \left( {C^{2} ,C^{3} } \right) - \mu \left( {C^{3} ,C^{4} } \right) \le 0 \hfill \\ \mu \left( {C^{2} } \right) + \mu \left( {C^{3} } \right) + \mu \left( {C^{4} } \right) - \mu \left( {C^{2} ,C^{4} } \right) - \mu \left( {C^{3} ,C^{4} } \right) \le 0 \hfill \\ \mu \left( {C^{1} ,C^{2} } \right) + \mu \left( {C^{1} ,C^{3} } \right) + \mu \left( {C^{1} ,C^{4} } \right) + \mu \left( {C^{2} ,C^{3} } \right) + \mu \left( {C^{2} ,C^{4} } \right) \hfill \\ + \mu \left( {C^{3} ,C^{4} } \right) - 2(\mu \left( {C^{1} } \right) + \mu \left( {C^{2} } \right) + \mu \left( {C^{3} } \right) + \mu \left( {C^{4} } \right)) = 1 \hfill \\ \mu \left( {C^{1} } \right) \in [0.15,0.2],\mu \left( {C^{2} } \right) \in [0.2,0.3],\mu \left( {C^{3} } \right) \in [0.2,0.25],\mu \left( {C^{4} } \right) \in [0.35,0.4] \hfill \\ \end{gathered} \right.} \\ \end{array} \hfill \\ \end{gathered} $$

Solving the above model using MATLAB, we obtain

$$ \begin{gathered} \mu (C^{1} {\text{) = 0}}{\text{.15,}}\quad \mu (C^{2} {\text{) = 0}}{\text{.2}},\quad \mu (C^{3} {\text{) = 0}}{\text{.2,}}\quad \mu (C^{4} {\text{) = 0}}{\text{.35}},\quad {\text{ }}\mu (C^{1} {\text{,C}}^{2} {\text{) = 0}}{\text{.35}}, \hfill \\ \mu (C^{1} {\text{,C}}^{3} {\text{) = 0}}{\text{.35}},\quad \mu (C^{1} {\text{,C}}^{4} {\text{) = 0}}{\text{.5}},\quad \mu (C^{2} {\text{,C}}^{3} {\text{)}} = {\text{0}}{\text{.4}},\quad \mu (C^{2} {\text{,C}}^{4} {\text{)}} = {\text{0}}{\text{.65}},\quad \mu (C^{3} {\text{,C}}^{4} {\text{)}} = {\text{0}}{\text{.55}}. \hfill \\ \end{gathered} $$

By Eq. (8), we obtain the four first-class evaluation criteria’ Shapley values as follows:

$$ \Phi _{{C^{1} }} (\mu ,C) = 0.2,\quad \Phi _{{C^{2} }} (\mu ,C) = 0.25,\quad {\text{ }}\Phi _{{C^{3} }} (\mu ,C) = 0.2,\quad {\text{ }}\Phi _{{C^{4} }} (\mu ,C) = 0.35. $$

On the basis of the Shapley values of the first-class criteria and sub-criteria, we obtain the synthetic weights of the sub-criteria as shown in Table 10.

Table 10 The weights of first-class criteria and sub-criteria
Step 6:

Calculate the net flow of each alternative based on the synthetic weights of sub-criteria, where

$$\phi (a_{1} ) = - 1.17, \quad \phi (a_{2} ) = - 0.42, \quad \phi (a_{3} ) = - 3.49, \quad \phi (a_{4} ) = - 2.30.$$
Step 7:

According to the net flow of each alternative, the ranking of the alternatives is \(a_{2} \succ a_{1} \succ a_{4} \succ a_{3}\) and the solid hardwood flooring \(a_{2}\) is the best choice.

However, this conclusion is obtained by the new score function for \(\rho = 0.5\). With respect to the different values of ρ, the ranking orders are obtained as shown in Table 11.

Table 11 Ranking orders using the new score function for the different values of ρ

From Table 11, one can see that the same ranking order is obtained with the different values of ρ. This means the ranking results are insensitive to the values of ρ.

Comparative analysis

It is necessary to compare the new method with other existing methods to validate the effectiveness and applicability of the new method for the selection of sustainable building materials. The compared methods include Chen et al.’s method (Chen et al. 2015) based on the LIFWA operator, Ou et al.’s method (Ou et al. 2018) based on the linguistic intuitionistic fuzzy TOPSIS method, and Liu and Shen’s method (Liu and Shen 2019) based on the linguistic intuitionistic fuzzy Choquet-an acronym in Portuguese for iterative multi-criteria decision making (C-TODIM) method. Note that none of these previous methods can cope with the situation where the weight information is incompletely known. To compare the new method with them, we here adopt two procedures: one assumes that all criteria have equal importance and the other adopts the weight information obtained by the new method. The related results and ranking orders are obtained as shown in Table 12.

Table 12 Ranking results with respect to different methods

From Table 12, one can observe that the same ranking results are obtained according to the new method and Liu and Shen’s method (Liu and Shen 2019), where \(a_{2} \succ a_{1} \succ a_{4} \succ a_{3}\). However, the difference between the final ranking values obtained by the new method is bigger than that derived from Liu and Shen’s method. Furthermore, the ranking results by Chen et al.’s method (Chen et al. 2015) are the same as those obtained by Ou et al.’s method (Ou et al. 2018). It is worth noting that the LIFWA operator used in Chen et al.’s method and Ou et al.’s method is unreasonable. When there exists \(s_{0}\) in the non-preferred degree of LIFNs, the final comprehensive non-preferred degree will be equal to \(s_{0}\) regardless of the values of the non-preferred degrees of other LIFNs. According to the above results of comparison analyses, there are some non-negligible distinctions among these methods:

  1. 1.

    None of these previous methods (Chen et al. 2015; Ou et al. 2018; Liu and Shen 2019) fully consider the interrelationships between criteria and deal with the situation where the weighting information is incompletely known;

  2. 2.

    Methods in (Chen et al. 2015; Ou et al. 2018) do not consider the DMs’ risk preference;

  3. 3.

    Method in (Liu and Shen 2019) uses λ-fuzzy measure to describe the interrelationship between criteria, which cannot cope with the complementary, redundant and independent interactions among elements in a set simultaneously.

Compared with the previous methods, the new method has three main advantages:

  1. 1.

    The new accuracy and score functions are defined, which consider the attitudes of the DMs;

  2. 2.

    A linguistic intuitionistic fuzzy PROMETHEE and PT based hybrid method is proposed, which considers the risk preference of the DMs;

  3. 3.

    New linguistic intuitionistic fuzzy entropy and similarity measure are defined, which are used to solve decision making with incomplete and interactive weighting information.

To compare the new method with previous methods more clearly, please see Table 13.

Table 13 Comparison of different methods

Conclusion

The selection of sustainable building materials plays an essential role in the sustainable construction and human health. Considering this problem, this paper first establishes the hierarchical evaluation structure including four first-class criteria such as economy, environment, society and technology. In addition, to facilitate the judgments for the sustainable building materials, we apply LIFNs to denote the qualitative preferred and non-preferred judgments simultaneously. Furthermore, we propose a MCDM method, which can deal with the interactive relationships among criteria under linguistic intuitionistic fuzzy environment. At first, a new accuracy function and a new score function are defined to rank LIFNs efficiently and reasonably, which considers the attitude of the DMs. Then, to cope with the situation where the weighting information is incompletely known and interactive, a model for deriving the fuzzy measure on criteria set is constructed, which is based on the new similarity measure and the Shapley function with respect to 2-additive measure. Finally, a linguistic intuitionistic fuzzy PROMETHEE and PT based hybrid method is given to obtain comprehensive criteria values and to reflect the risk aversion psychology of DMs. To demonstrate the feasibility and efficiency of the new method, a practical MCDM problem about sustainable indoor flooring material selection is studied. Nevertheless, the new method is more complex than some previous ones. Furthermore, it does not study the situations where the interaction relationships among criteria are known before determining their weights. In application aspect, it mainly discusses the selection of sustainable building material.

This paper focuses on a linguistic intuitionistic fuzzy PROMETHEE method based on similarity measure. Similarly, we can study LIFNs using other information measures, such as distance measure. Furthermore, we shall further study other linguistic intuitionistic fuzzy MCDM methods, such as linguistic intuitionistic fuzzy VIKOR method, linguistic intuitionistic fuzzy ELimination and Choice Expressing Reality (ELECTRE) method and linguistic intuitionistic fuzzy TODIM method. In addition, the new method can be extended to large group decision making as well as decision making with multi-granularity linguistic term sets. Besides the theoretical aspect, this paper mainly discusses the sustainable building material selection, and it is necessary to apply this new method to solve some other practical decision-making problems, such as supply chain management (Chen et al. 2015), staff evaluation (Liu and Shen 2019), coal mine safety evaluation (Zhang et al. 2017) and project management (Meng et al. 2020).

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Acknowledgements

The authors gratefully thank the Editor-in-Chief Professor Vincenzo Loia, the Associated Editor Professor Aniello Castiglione, and three anonymous referees for their valuable and constructive comments, which have much improved the paper. This work was supported by the Major Project for National Natural Science Foundation of China (no. 72091515).

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Meng, F., Dong, B. Linguistic intuitionistic fuzzy PROMETHEE method based on similarity measure for the selection of sustainable building materials. J Ambient Intell Human Comput (2021). https://doi.org/10.1007/s12652-021-03338-y

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Keywords

  • Sustainable building material
  • PROMETHEE method
  • Prospect theory
  • Linguistic intuitionistic fuzzy number
  • The Shapley function