1 Introduction

Moore [25] defined the concept of interval-valued fuzzy numbers (IVFNs) and their extended operational laws, which are of great importance to represent the statuses of objects or the evaluations of parameters associated with real-world problems, for example, the lowest and highest price of a stock, daily temperatures of a region, blood pressure of a human being, etc. Wang and Li [36] and Hong and Lee [16] studied the correlation coefficient and distance measure of IVFNs, respectively. Since the IVFN was developed, it has been widely studied and applied to many real applications, for example, fuzzy metrics [48], learning and reasoning fields [14, 22], and forecasting [9, 15, 24]. The application of IVFNs to decision making includes economics, engineering, and management sciences [10, 25, 30, 39–42, 50, 53, 56].

In many real-world problems with IVFNs, the comparisons among IVFNs play an important role. Different kinds of techniques have been developed to compare IVFNs since the work finished by Ishibuchi and Tanaka [19], in which five-order relations for comparing and ranking intervals based on the comparison of the lower and upper values or the comparison of the center and radius values were introduced. Okada and Gen [27] and Chanas and Kuchta [7, 8] extended the work given by Ishibuchi and Tanaka [19] and proposed some new approaches to rank IVFNs, which can reduce the number of incomparable pairs or simplified the initial comparing operations. Sengupta et al. [29] and Sengupta and Pal [31] developed two ranking schemes named acceptability index and fuzzy preference ordering for intervals, respectively. Sengupta and Pal [30] summarized these methods and named them deterministic comparable methods.

For the reason that there exist incomparable cases when using deterministic comparable methods, the concepts of possibility degrees were developed. Nakahara et al. [26] defined the probability of the interval inequality for intervals using the concept of probabilistic. Since then, Kundu [23], Facchinetti et al. [12], Jahanshahloo et al. [20], Jiang et al. [21], Senguta and Pal [31], Wang et al. [37], and Sun and Yao [35] extended types of possibility degrees and applied them to group decision making, systems’ reliability, and optimization issues.

Recently, Song et al. [33] proposed a two-grade approach to rank interval data, in which an entire dominance degree was used in the first grade and an entire directional distance index was utilized to handle the cases that cannot be ranked in the first grade. By using an axiomatic set of membership, non-membership, vague and precise score functions of an interval-valued intuitionistic fuzzy numbers, Sivaraman et al. [32] introduced a new method for the complete ranking of incomplete interval information.

However, these existing methodologies for comparing and ranking interval numbers are all shown in certain degree with crisp format. Recently, Bodenhofer [5, 6] considered that it is a serious restriction for ranking fuzzy sets based on strictly crisp comparisons. Bodenhofer [5] introduced a general framework for comparing fuzzy sets with respect to fuzzy ordering in a gradual way. Other studies on fuzzy ordering of fuzzy sets could be seen in Ovchinnikov [28], Horiuchi and Tamura [17], Yoshida and Kerre [46], Zadeh [47], Stamenković et al. [34], Ignjatović et al. [18], and Zhang et al. [49]. But these schemes were all considered to fuzzify crisp orderings. It is worth noting that a direct fuzzy way to deal with interval ranking issues would be a new direction.

In this paper, we are going to study on the development of the interval comparing methodologies originating from the work of Moore [25]. Intuitionistic fuzzy sets [2, 3] are introduced to model vagueness and intuitive uncertain in real world by admitting multiple membership degrees, which are usually used to describe one’s preference of one object to another object [50–52, 61–62] and the performance of one object with respect to given attribute [55–59]. Correspondingly, in the interval comparing and ranking problems, there exist multiple cases, for instance, incomparable cases of nested interval numbers and the intuitive inconsistency starting from the cases of ranking equi-centered interval numbers [19], i.e., the two special cases that \(\tilde{a} = [a - \nabla ,\;b + \nabla ]\), \(a > \nabla\), \(\tilde{b} = [a,\;b],\) and \(\tilde{a}_{i} = [a_{i} ,\;b_{i} ],\;i = 1,\;2\), \(a_{1} + b_{1} = a_{2} + b_{2}\) are incomparable according to existed comparison laws. Herein, we build an intuitionistic fuzzy ordering relation for interval numbers in order to deal with the above two questions.

For the reason that an interval number is composed by all possible numbers between the lower bound and the upper bound, it will be more accurate to compare any two intervals according to certain subintervals, i.e., a subinterval in an interval has certain relationship with other subintervals of another interval. In this paper, we will define an intuitionistic fuzzy possibility degree to compare two interval numbers, in which the comparison between two interval numbers is composed by three measures, i.e., the degree that an interval is absolutely greater than the other one, the degree that an interval is absolutely less than the other one, and the degree that an interval is incomparable with the other one.

The rest of this paper is organized as follows. In Sect. 2, we recall some of the basic definitions and notations. The concept of intuitionistic fuzzy possibility degree (IFPD) for comparing of interval numbers is put forward in Sect. 3. Some of its properties are further studied in this section. However in Sect. 4, the proposed IFPD is applied to fuzzy multiple attributes decision making (FMADM). Finally, a case study of laptop’s selection issue is presented in Sect. 5.

2 Preliminaries

In this section, we briefly review the concepts of interval number, intuitionistic fuzzy sets, and their related operation laws.

Definition 1

(Moore [25]). \(\tilde{a}\) is called an interval number if

$$\tilde{a} = [a^{ - } ,\;\;a^{ + } ] = \{ x|a^{ - } \le x \le a^{ + } ,\;\;x \in R\},$$

where \(a^{ - } ,\;\;a^{ + }\) are the left and right limit of the interval \(\tilde{a}\) on the real line \(R\), respectively. If \(a^{ - } = a^{ + } = a\), then \(\tilde{a}\) degenerates to be a real number.

Moore [25] noted that an interval number \(\tilde{a}\) can also be written as \(\tilde{a} = (m_{{\tilde{a}}} ,\;w_{{\tilde{a}}} )\), where \(m_{{\tilde{a}}} = {{(a^{ - } + a^{ + } )} \mathord{\left/ {\vphantom {{(a^{ - } + a^{ + } )} 2}} \right. \kern-0pt} 2}\) and \(w_{{\tilde{a}}} = {{(a^{ + } - a^{ - } )} \mathord{\left/ {\vphantom {{(a^{ + } - a^{ - } )} 2}} \right. \kern-0pt} 2}\) represent the midpoint and width of \(\tilde{a}\), respectively.

For any two interval numbers \(\tilde{a}\) = [a −, a +] and \(\tilde{b}\) = [b −, b +], \(\lambda \ge 0\) is a positive scalar, the following operational laws are valid (Moore [25]; Sengupta and Pal [30]):

  1. (1)

    \(\tilde{a} + \tilde{b} = [a^{ - } + b^{ - } ,\;\;a^{ + } + b^{ + } ]\);

  2. (2)

    \(\tilde{a} - \tilde{b} = [a^{ - } - b^{ + } ,\;\;a^{ + } - b^{ - } ]\);

  3. (3)

    \(\lambda \cdot \tilde{a} = [\lambda a^{ - } ,\;\;\lambda a^{ + } ]\);

  4. (4)

    \({1 \mathord{\left/ {\vphantom {1 {\tilde{a}}}} \right. \kern-0pt} {\tilde{a}}} = [{1 \mathord{\left/ {\vphantom {1 {a^{ + } }}} \right. \kern-0pt} {a^{ + } }},\;\;{1 \mathord{\left/ {\vphantom {1 {a^{ - } }}} \right. \kern-0pt} {a^{ - } }}]\) (\(a^{ - } > 0\)).

Definition 2

(Atanassov [2, 3]). An intuitionistic fuzzy set (IFS) A defined on an universe X is given according to

$${\varvec{A}} = \left\{ {\left\langle {x,\mu _{A} (x),v_{A} (x)} \right\rangle \left| {x \in {\varvec{X}}} \right.} \right.\},$$

where \(\mu_{A} :\varvec{X} \to [0,\;\;1]\) and \(\nu_{A} :\varvec{X} \to [0,\;\;1]\) denote the degree of membership and degree of non-membership of x to A, respectively, and for all \(x \in \varvec{X}\), the two degrees satisfy the condition that \(0 \le \mu_{A} (x) + \nu_{A} (x) \le 1\). Besides, the intuitionistic index of an element \(x \in \varvec{X}\) in A with the following expression

$$\pi_{A} (x) = 1 - \mu_{A} (x) - \nu_{A} (x)$$

is called the hesitancy degree of x to A.

The complementary set of A, which can be denoted by A c, is defined according to

$$\varvec{A}^{c} = \left\{ {\left\langle {x,v_{A} (x),\mu _{A} (x)} \right\rangle \big| {x \in \varvec{X}} } \right\}.$$

For convenience, Xu [44] called \(\alpha = \left\langle {\mu _{\alpha } ,v_{\alpha} } \right\rangle\) an intuitionistic fuzzy number (IFN).

Distance is an effective measure to describe the difference between two IFNs. Szmidt and Kacprzyk [54] defined the following distance measure:

Definition 3

([54]). Assume that \(\alpha = \left\langle {\mu _{\alpha } ,v_{\alpha } ,\pi _{\alpha } } \right\rangle\) and \(\beta = \left\langle {\mu_{\beta } ,\;\;\nu_{\beta } ,\;\pi_{\beta } } \right\rangle\) are any two IFNs, then the distance between \(\alpha\) and \(\beta\) can be calculated by

$$d(\alpha ,\;\;\beta ) = \frac{1}{2}\left( {\left| {\mu_{\alpha } - \mu_{\beta } } \right| + \left| {\nu_{\alpha } - \nu_{\beta } } \right| + \left| {\pi_{\alpha } - \pi_{\beta } } \right|} \right)$$

According to Definition 3, d(α, β = 0) if and only if \(\mu_{\alpha } = \mu_{\beta }\) and \(v_{\alpha } = v_{\beta }\), i.e., \(\pi_{\alpha } = \pi_{\beta }\), which is equivalent to \(\alpha = \beta\).

3 Intuitionistic Fuzzy Rankings of Interval-Valued Fuzzy Numbers

3.1 A Geometrical Illustration for Ranking IVFNs

For any two compared interval numbers \(\tilde{a} = [a^{ - } ,\;\;a^{ + } ]\) and \(\tilde{b} = [b^{ - } ,\;\;b^{ + } ]\), their locations on the number axis can be mainly shown in Fig. 1.

Fig. 1
figure 1

Locations of compared intervals on number axis

Full size image

Because \(\tilde{a}\) and \(\tilde{b}\) are symmetrical, i.e., (R1)–(R3) in Fig. 1 are corresponding cases when interchanging \(\tilde{a}\) and \(\tilde{b}\) in (L1)–(L3), we mainly analyze the latter ones.

Case L1

\(I_{1} = \tilde{a},\;I_{2} = \tilde{b}\), for any \(x \in I_{1}\) and \(y \in I_{2}\) (\(x\) and \(y\) are real numbers), we have \(x \ge y\), i.e., \(\tilde{a}\) is greater than \(\tilde{b}\);

Case L2

\(II_{1} \bigcup {II_{2} } = \tilde{a},II_{1} \bigcup {II_{3} } = \tilde{b},II_{1} = [a^{ - } ,b^{ + } ],II_{2} = [b^{ - } ,a^{ + } ],II_{3} = [b^{ - } ,a^{ - } ]\) and \(II_{1} = \tilde{a} \bigcap \tilde{b}\), we have

  1. (1)

    For any \(x \in II_{2} \subseteq \tilde{a}\) or \(y \in II_{3} \subseteq \tilde{b}\), \(\tilde{a}\) is greater than \(\tilde{b}\);

  2. (2)

    If \(x,\;y \in II_{1} = \tilde{a} \cap \tilde{b}\), then \(\tilde{a}\) and \(\tilde{b}\) are incomparable.

Case L3

\(III_{1} \bigcup {III_{2} \bigcup {III_{3} } } = \tilde{a},III_{2} = \tilde{b},III_{1} = [a^{ - } ,b^{ - } ],III_{2} = [b^{ - } ,b^{ + } ],III_{3} = [b^{ + } ,a^{ + } ]\) and \(III_{2} = \tilde{a} \cap \tilde{b}\), then we have

  1. (1)

    For any \(x \in III_{1} \subseteq \tilde{a}\) and \(y \in III_{2} = \tilde{b}\), \(\tilde{a}\) is absolutely less than \(\tilde{b}\);

  2. (2)

    For any \(x \in III_{3} \subseteq \tilde{a}\) and \(y \in III_{2} = \tilde{b}\), \(\tilde{a}\) is absolutely greater than \(\tilde{b}\);

  3. (3)

    If \(x,\;y \in III_{2} = \tilde{a} \cap \tilde{b}\), then \(\tilde{a}\) and \(\tilde{b}\) are incomparable.

From Case L1 to Case L3, the comparison between two interval numbers can be partitioned as three aspects, i.e., an interval is absolutely greater than another interval, an interval is absolutely less than another interval, and an interval is incomparable with another interval.

Thus, it can be derived that a comparison by analyzing different situations would be more reasonable to rank intervals.

3.2 Intuitionistic Fuzzy Order of IVFNs

As mentioned above, in this subsection, we will define an intuitionistic fuzzy possibility degree for comparing any two IVFNs.

For convenience, let \(\Omega\) be the set of all IVFNs.

Definition 4

Assume that \(\tilde{a}, \tilde{b} \in \Omega\), where \(\tilde{a} = [a^{ - } ,\;\;a^{ + } ]\) and \(\tilde{b} = [b^{ - } ,\;\;b^{ + } ]\). The intuitionistic fuzzy possibility degree of \(\tilde{a} \ge \tilde{b}\), which is denoted by \(p(\tilde{a} \ge \tilde{b})\), is defined as follows

$$p(\tilde{a} \ge \tilde{b}) = \left\langle {\mu_{{p(\tilde{a} \ge \tilde{b})}} ,\nu_{{p(\tilde{a} \ge \tilde{b})}} ,\pi_{{p(\tilde{a} \ge \tilde{b})}} } \right\rangle ,$$
(1)

where

$$\mu_{{p\left( {\tilde{a} \ge \tilde{b}} \right)}} = \frac{{\left| \varvec{X} \right| + \left| \varvec{Y} \right|}}{{\left| {\tilde{a}} \right| + \left| {\tilde{b}} \right| - \left| {\tilde{a} \cap \tilde{b}} \right|}},\;\;\nu_{{p\left( {\tilde{a} \ge \tilde{b}} \right)}} = \frac{{\left| {\neg \varvec{X}} \right| + \left| {\neg \varvec{Y}} \right|}}{{\left| {\tilde{a}} \right| + \left| {\tilde{b}} \right| - \left| {\tilde{a} \cap \tilde{b}} \right|}},\;\;\pi_{{p\left( {\tilde{a} \ge \tilde{b}} \right)}} = \frac{{\left| {\tilde{a} \cap \tilde{b}} \right|}}{{\left| {\tilde{a}} \right| + \left| {\tilde{b}} \right| - \left| {\tilde{a} \cap \tilde{b}} \right|}} ,$$
(2)

\(\varvec{X} = \{ x|x \in \tilde{a},\forall \;y \in \tilde{b},x \ge y\} ,\;\varvec{Y} = \{ y|y \in \tilde{b},\forall \;x \in \tilde{a},y \le x\}\) and \(\neg \varvec{X},\;\neg \varvec{Y}\) are logical negations of \(\varvec{X},\;\varvec{Y}\), then \(\mu_{{p(\tilde{a},\tilde{b})}}\) is called the possibility degree that \(\tilde{a}\) is absolutely greater than \(\tilde{b}\), \(\nu_{{p(\tilde{a},\tilde{b})}}\) is called the possibility degree that \(\tilde{a}\) is absolutely less than \(\tilde{b}\), and \(\pi_{{p(\tilde{a},\tilde{b})}}\) is called the hesitant possibility degree for the comparison of \(\tilde{a}\) and \(\tilde{b}\).

According to Definition 4, given that \(\tilde{a}, \tilde{b} \in \Omega\), where \(\tilde{a} = [a^{ - } ,\;\;a^{ + } ]\) and \(\tilde{b} = [b^{ - } ,\;\;b^{ + } ]\), then Cases L1 to Case L3 can be detailed as follows:

Case L1

If \(a^{ - } \ge b^{ + }\), we have \(\tilde{a} \bigcap \tilde{b}\,=\,\)Ø, \(\varvec{X = }\tilde{a}\), \(\varvec{Y = }\tilde{b},\,\neg\,\varvec{X}\,=\,\)Ø, \(\neg\,\varvec{Y\,=\,}\) Ø, then

$$\left| {\tilde{a} \cap \tilde{b}} \right| = 0,\,\left| \varvec{X} \right|\varvec{ = }\left| {\tilde{a}} \right|,\,\left| \varvec{Y} \right|\varvec{ = }\left| {\tilde{b}} \right|,\,\left| {\neg \varvec{X}} \right| = 0,\,\left| {\neg \varvec{Y}} \right|\varvec{ = }0.$$

According to Eq. (2),

$$\mu _{{p(\tilde{a} \ge \tilde{b})}} = 1,v_{{p(\tilde{a} \ge \tilde{b})}} = 0,\pi _{{p(\tilde{a} \ge \tilde{b})}} = 0,\,{\text{i}}{\text{.e}}{\text{.,}}\,p(\tilde{a} \ge \tilde{b}) = \left\langle {1,0,0} \right\rangle .$$

Case L2

If \(b^{ - } \le a^{ - } \le b^{ + } \le a^{ + }\), then,

$$\left| {\tilde{a}} \right| = a^{ + } - a^{ - },\,\left| {\tilde{b}} \right| = b^{ + } - b^{ - },\,\left| {\tilde{a} \cap \tilde{b}} \right| = b^{ + } - a^{ - },\,\left| \varvec{X} \right|\varvec{ = }a^{ + } - b^{ + },\,\left| \varvec{Y} \right|\varvec{ = }a^{ - } - b^{ - },\,\left| {\neg \varvec{X}} \right| = 0,\,\left| {\neg \varvec{Y}} \right|\varvec{ = }0 .$$

According to Eq. (2),

$$\mu _{{p(\tilde{a} \ge \tilde{b})}} = \frac{{\left| {\varvec{X}} \right| + \left| Y \right|}}{{\left| {\tilde{a}} \right| + \left| {\tilde{b}} \right| - \left| {\tilde{a}\bigcap {\tilde{b}} } \right|}} = \frac{{(a^{ + } - b^{ + } ) + (a^{ - } - b^{ - } )}}{{(a^{ + } - a^{ - } ) + (b^{ + } - b^{ - } ) - (b^{ + } - a^{ - } )}} = \frac{{a^{ + } - b^{ + } + a^{ - } - b^{ - } }}{{a^{ + } - b^{ - } }}.$$

Similarly, \(v_{{p(\tilde{a} \ge \tilde{b})}} = 0,\pi _{{p(\tilde{a} \ge \tilde{b})}} = \frac{{b^{ + } - a^{ - } }}{{a^{ + } - b^{ - } ,}}\) i.e.,

$$p(\tilde{a} \ge \tilde{b}) = \left\langle {\frac{{a^{ + } - b^{ + } + a^{ - } - b^{ - } }}{{a^{ + } - b^{ - } }},0,\frac{{b^{ + } - a^{ - } }}{{a^{ + } - b^{ - } }}} \right\rangle .$$

Case L3

If \(a^{ - } \le b^{ - } \le b^{ + } \le a^{ + }\), then

$$\left| {\tilde{a}} \right| = a^{ + } - a^{ - } ,\,\left| {\tilde{b}} \right| = b^{ + } - b^{ - } ,\,\left| {\tilde{a} \cap \tilde{b}} \right| = b^{ + } - b^{ - },\,\left| \varvec{X} \right|\varvec{ = }a^{ + } - b^{ + },\,\left| \varvec{Y} \right|\varvec{ = }0,\,\left| {\neg \varvec{X}} \right| = b^{ - } - a^{ - },\,\left| {\neg \varvec{Y}} \right|\varvec{ = }0.$$

According to Eq. (2),

$$\mu _{{p(\tilde{a} \ge \tilde{b})}} = \frac{{\left| {\varvec{X}} \right| + \left| {\varvec{X}} \right|}}{{\left| {\tilde{a}} \right| + \left| {\tilde{b}} \right| - \left| {\tilde{a}\bigcap {\tilde{b}} } \right|}} = \frac{{(a^{ + } - b^{ + } ) + 0}}{{(a^{ + } - a^{ - } ) + (b^{ + } - b^{ - } ) - (b^{ + } - b^{ - } )}} = \frac{{a^{ + } - b^{ + } }}{{a^{ + } - a^{ - } }}.$$

Similarly, \(v_{{p(\tilde{a} \ge \tilde{b})}} = \frac{{\left| X \right| + \left| Y \right|}}{{\left| {\tilde{a}} \right| + \left| {\tilde{b}} \right| - \left| {\tilde{a}\bigcap {\tilde{b}} } \right|}} = \frac{{b^{ - } - a^{ - } }}{{a^{ + } - a^{ - } }},\pi _{{p(\tilde{a} \ge \tilde{b})}} = \frac{{b^{ + } - b^{ - } }}{{a^{ + } - a^{ - } }}{\text{i}}{\text{.e}}{\text{.,}}\)

$$p(\tilde{a} \ge \tilde{b}) = \left\langle {\frac{{a^{ + } - b^{ + } }}{{a^{ + } - a^{ - } }},\;\;\frac{{b^{ - } - a^{ - } }}{{a^{ + } - a^{ - } }},\;\;\frac{{b^{ + } - b^{ - } }}{{a^{ + } - a^{ - } }}} \right\rangle.$$

Thus, Eq. (2) can be summarized as follows:

$$p(\tilde{a} \ge \tilde{b}) = \left\{ {\begin{array}{*{20}l} {\left\langle {1,0,0} \right\rangle ,} &\quad {if\,a^{ - } \ge b^{ + } } \\ {\left\langle {\frac{{a^{ + } - b^{ + } + a^{ - } - b^{ - } }}{{a^{ + } - b^{ - } }},0,\frac{{b^{ + } - a^{ - } }}{{a^{ + } - b^{ - } }}} \right\rangle ,} &\quad {if\,b^{ - } \le a^{ - } \le b^{ + } \le a^{ + } } \\ {\left\langle {\frac{{a^{ + } - b^{ + } }}{{a^{ + } - a^{ - } }},\frac{{b^{ - } - a^{ - } }}{{a^{ + } - a^{ - } }},\frac{{b^{ + } - b^{ - } }}{{a^{ + } - a^{ - } }}} \right\rangle ,} &\quad {if\,a^{ - } \le b^{ - } \le b^{ + } \le a^{ + } } \\ \end{array} } \right..$$
(3)

According to Eq. (3), it is obvious that the following properties of intuitionistic fuzzy possibility degree of \(p(\tilde{a} \ge \tilde{b})\) are valid:

Property 1

\(\mu_{{p(\tilde{a} \ge \tilde{b})}} ,\;\nu_{{p(\tilde{a} \ge \tilde{b})}} ,\;\pi_{{p(\tilde{a} \ge \tilde{b})}} \in [0,1]\) and \(\mu_{{p(\tilde{a} \ge \tilde{b})}} + \nu_{{p(\tilde{a} \ge \tilde{b})}} + \pi_{{p(\tilde{a} \ge \tilde{b})}} = 1\).

Especially, if \(\pi_{{p(\tilde{a} \ge \tilde{b})}} = 0\), then \(\mu_{{p(\tilde{a} \ge \tilde{b})}} + \nu_{{p(\tilde{a} \ge \tilde{b})}} = 1\).

Property 2

\(\mu_{{p(\tilde{b} \ge \tilde{a})}} = \nu_{{p(\tilde{a} \ge \tilde{b})}} ,\;\nu_{{p(\tilde{b} \ge \tilde{a})}} = \mu_{{p(\tilde{a} \ge \tilde{b})}}\), \(\pi_{{p(\tilde{b} \ge \tilde{a})}} = \pi_{{p(\tilde{a} \ge \tilde{b})}}\).

Property 3

If \(\tilde{a} = \tilde{b}\), then \(p(\tilde{a} \ge \tilde{b}) = \left\langle {0,\;0,\;1} \right\rangle\), i.e., \(\tilde{a}\) is equivalent to itself with 100 percent.

In order to make a comparison between \(\tilde{a}\) and \(\tilde{b}\), the meaning of \(p(\tilde{a} \ge \tilde{b})\) is given by Definition 5.

Definition 5

Assume that \(\Omega\) is the set of all IVFNs and \(\tilde{a},\;\tilde{b} \in \varOmega\), where \(\tilde{a} = [a^{ - } ,\;\;a^{ + } ]\) and \(\tilde{b} = [b^{ - } ,\;\;b^{ + } ]\), \(p(\tilde{a} \ge \tilde{b})\) is the intuitionistic fuzzy possibility degree of \(\tilde{a} \ge \tilde{b}\), then

  1. (1)

    If \(\mu_{{p(\tilde{a} \ge \tilde{b})}} - \nu_{{p(\tilde{a} \ge \tilde{b})}} > 0\), then \(\tilde{b}\) is called to be dominated by \(\tilde{a}\) with certain degree \(\mu_{{p(\tilde{a} \ge \tilde{b})}}\), denoted by \(\tilde{a} >_{{\mu_{{p(\tilde{a} \ge \tilde{b})}} }} \tilde{b}\);

  2. (2)

    If \(\mu_{{p(\tilde{a} \ge \tilde{b})}} - \nu_{{p(\tilde{a} \ge \tilde{b})}} < 0\), then \(\tilde{a}\) is said to be dominated by \(\tilde{b}\) with certain degree \(v_{{p(\tilde{a} \ge \tilde{b})}}\), denoted by \(\tilde{b} >_{{v_{{p(\tilde{a} \ge \tilde{b})}} }} \tilde{a}\);

  3. (3)

    If \(\mu_{{p(\tilde{a} \ge \tilde{b})}} - \nu_{{p(\tilde{a} \ge \tilde{b})}} = 0\), then \(\tilde{a}\) is equivalent to \(\tilde{b}\) with possibility degree \(\pi_{{p(\tilde{a},\tilde{b})}}\), denoted by \(\tilde{a}\sim_{{\pi_{{p(\tilde{a} \ge \tilde{b})}} }} \tilde{b}\).

Theorem 1

Suppose that \(\tilde{a},\,\tilde{b},\,\tilde{c} \in \Omega\), the intuitionistic fuzzy possibility degree \(p( \cdot )\) satisfies the following properties:

  1. (1)

    Bounded: \(\left\langle {0,1,0} \right\rangle \le p(\tilde{a} \ge \tilde{b}) \le \left\langle {1,0,0} \right\rangle\);

  2. (2)

    Transitivity: If \(\tilde{a} >_{{\mu_{{p(\tilde{a} \ge \tilde{b})}} }} \tilde{b}\) and \(\tilde{b} >_{{\mu_{{p(\tilde{b} \ge \tilde{c})}} }} \tilde{c}\), then \(\tilde{a} >_{{\mu_{{p(\tilde{a} \ge \tilde{c})}} }} \tilde{c}\).

Proof

  1. (1)

    According to Property 1, \(\left\langle {0,1,0} \right\rangle \le p(\tilde{a} \ge \tilde{b}) \le \left\langle {1,0,0} \right\rangle\) is natural.

  2. (2)

    If \(\tilde{a} >_{{\mu_{{p(\tilde{a} \ge \tilde{b})}} }} \tilde{b}\) and \(\tilde{b} >_{{\mu_{{p(\tilde{b} \ge \tilde{c})}} }} \tilde{c}\), according to Eq. (3) and Definition 5, there are nine cases, which can be proved as below:

Case 1

If \(\mu_{{p(\tilde{a} \ge \tilde{b})}} = 1\) and \(\mu_{{p(\tilde{b} \ge \tilde{c})}} = 1\), i.e., \(a^{ - } \ge b^{ + } \ge b^{ - } \ge c^{ + }\), then \(\mu_{{p(\tilde{a} \ge \tilde{c})}} = 1\), i.e., \(\tilde{a} >_{1} \tilde{c}\);

Case 2

If \(\mu_{{p(\tilde{a} \ge \tilde{b})}} = 1\) and \(\mu_{{p(\tilde{b} \ge \tilde{c})}} - v_{{p(\tilde{b} \ge \tilde{c})}} > 0\) with the second case in Eq. (3), i.e., \(a^{ - } \ge b^{ + }\) and \(c^{ - } \le b^{ - } \le c^{ + } \le b^{ + }\), since \(a^{ - } \ge b^{ + } \ge c^{ + }\), thus \(\mu_{{p(\tilde{a} \ge \tilde{c})}} = 1\), i.e., \(\tilde{a} >_{1} \tilde{c}\);

Case 3

If \(\mu_{{p(\tilde{a} \ge \tilde{b})}} = 1\) and \(\mu_{{p(\tilde{b} \ge \tilde{c})}} - v_{{p(\tilde{b} \ge \tilde{c})}} > 0\) with the third case in Eq. (3), i.e., \(a^{ - } \ge b^{ + }\) and \((c^{ + } - b^{ + } ) - (b^{ - } - c^{ - } ) > 0\), for the reason that \(b^{ - } \le c^{ - } \le c^{ + } \le b^{ + }\), thus \(a^{ - } \ge b^{ + } \ge c^{ + }\), we have \(\mu_{{p(\tilde{a} \ge \tilde{c})}} = 1\), i.e., \(\tilde{a} >_{1} \tilde{c}\);

Case 4

If \(\mu_{{p(\tilde{a} \ge \tilde{b})}} - v_{{p(\tilde{a} \ge \tilde{b})}} > 0\) with the second case in Eq. (3) and \(\mu_{{p(\tilde{b} \ge \tilde{c})}} = 1\), i.e., \(b^{ - } \le a^{ - } \le b^{ + } \le a^{ + }\) and \(b^{ - } \ge c^{ + }\), thus \(a^{ - } \ge b^{ - } \ge c^{ + }\), we have \(\mu_{{p(\tilde{a} \ge \tilde{c})}} = 1\), i.e., \(\tilde{a} >_{1} \tilde{c}\);

Case 5

If both \(\mu_{{p(\tilde{a} \ge \tilde{b})}} - v_{{p(\tilde{a} \ge \tilde{b})}} > 0\) and \(\mu_{{p(\tilde{b} \ge \tilde{c})}} - v_{{p(\tilde{b} \ge \tilde{c})}} > 0\) hold with the second case in Eq. (3), i.e., \(b^{ - } \le a^{ - } \le b^{ + } \le a^{ + }\) and \(c^{ - } \le b^{ - } \le c^{ + } \le b^{ + }\), so \(a^{ + } \ge c^{ + }\) and \(c^{ - } \le a^{ - }\), besides, \(a^{ + } - b^{ + } + a^{ - } - b^{ - } > 0\) and \(b^{ + } - c^{ + } + b^{ - } - c^{ - } > 0\).

Thus, the following two situations are considered:

If c + ≤ a −, i.e., \(c^{ - } \le c^{ + } \le a^{ - }\), we have \(\mu_{{p(\tilde{a} \ge \tilde{c})}} = 1\), i.e., \(\tilde{a} >_{1} \tilde{c}\);

If c + ≤ a −, i.e., \(c^{ - } \le a^{ - } \le c^{ + } \le a^{ + }\), then \(\mu_{{p(\tilde{a} \ge \tilde{c})}} \ne 0\) and \(v_{{p(\tilde{a} \ge \tilde{c})}} = 0\), we have \(\mu_{{p(\tilde{a} \ge \tilde{c})}} - v_{{p(\tilde{a} \ge \tilde{c})}} > 0\), i.e., \(\tilde{a} >_{{\mu_{{p(\tilde{a} \ge \tilde{c})}} }} \tilde{c}\).

Case 6

If \(\mu_{{p(\tilde{a} \ge \tilde{b})}} - v_{{p(\tilde{a} \ge \tilde{b})}} > 0\) and \(\mu_{{p(\tilde{b} \ge \tilde{c})}} - v_{{p(\tilde{b} \ge \tilde{c})}} > 0\) hold with the second and the third case in Eq. (3), respectively, i.e., \(b^{ - } \le a^{ - } \le b^{ + } \le a^{ + }\) and \(b^{ - } \le c^{ - } \le c^{ + } \le b^{ + }\), so \(c^{ + } \le a^{ + }\), \(a^{ + } - b^{ + } + a^{ - } - b^{ - } > 0\) and \(b^{ + } - c^{ + } + b^{ - } - c^{ - } > 0\).

Thus, the following two situations are considered:

If c + ≤ a −, i.e., \(c^{ - } \le c^{ + } \le a^{ - }\), we have \(\mu_{{p(\tilde{a} \ge \tilde{c})}} = 1\), i.e., \(\tilde{a} >_{1} \tilde{c}\);

If c + ≥ a − and \(c^{ - } \le a^{ - }\), i.e., \(b^{ + } \le a^{ + }\) and \(b^{ - } \le a^{ - }\), because \(b^{ + } - c^{ + } + b^{ - } - c^{ - } > 0\), we have \(a^{ + } - c^{ + } + a^{ - } - c^{ - } > 0\), thus, \(\mu_{{p(\tilde{a} \ge \tilde{c})}} - v_{{p(\tilde{a} \ge \tilde{c})}} > 0\), i.e., \(\tilde{a} >_{{\mu_{{p(\tilde{a} \ge \tilde{c})}} }} \tilde{c}\);

If c − ≥ a −, since \(b^{ - } \le a^{ - } \le b^{ + } \le a^{ + }\) and \(b^{ + } - c^{ + } + b^{ - } - c^{ - } > 0\), we have \(a^{ + } - c^{ + } + a^{ - } - c^{ - } > 0\), thus \(\mu_{{p(\tilde{a} \ge \tilde{c})}} - v_{{p(\tilde{a} \ge \tilde{c})}} > 0\), i.e., \(\tilde{a} >_{{\mu_{{p(\tilde{a} \ge \tilde{c})}} }} \tilde{c}\).

Case 7

If \(\mu_{{p(\tilde{a} \ge \tilde{b})}} - v_{{p(\tilde{a} \ge \tilde{b})}} > 0\) with the third case in Eq. (3) and \(\mu_{{p(\tilde{b} \ge \tilde{c})}} = 1\), \(a^{ - } \le b^{ - } \le b^{ + } \le a^{ + }\) and \(b^{ - } \ge c^{ + }\), besides, \(a^{ + } - b^{ + } + a^{ - } - b^{ - } > 0\), then

If c + ≤ a −, then \(\mu_{{p(\tilde{a} \ge \tilde{c})}} = 1\), i.e., \(\tilde{a} >_{1} \tilde{c}\);

If c − ≤ a − ≤ c + ≤ a +, the conclusion that \(a^{ + } - c^{ + } + a^{ - } - c^{ - } > 0\) is obvious, thus \(\mu_{{p(\tilde{a} \ge \tilde{c})}} - v_{{p(\tilde{a} \ge \tilde{c})}} > 0\), i.e., \(\tilde{a} >_{{\mu_{{p(\tilde{a} \ge \tilde{c})}} }} \tilde{c}\);

If a − ≤ c − ≤ c + ≤ a −, for the reason that \(a^{ + } - b^{ + } + a^{ - } - b^{ - } > 0\) and \(b^{ - } \ge c^{ + }\), then \((a^{ + } - c^{ + } + a^{ - } - c^{ - } ) - (a^{ + } - b^{ + } + a^{ - } - b^{ - } ) = b^{ + } - c^{ + } + b^{ - } - c^{ - } > 0\), thus \(\mu_{{p(\tilde{a} \ge \tilde{c})}} - v_{{p(\tilde{a} \ge \tilde{c})}} > 0\), i.e., \(\tilde{a} >_{{\mu_{{p(\tilde{a} \ge \tilde{c})}} }} \tilde{c}\);

Case 8

If \(\mu_{{p(\tilde{a} \ge \tilde{b})}} - v_{{p(\tilde{a} \ge \tilde{b})}} > 0\) and \(\mu_{{p(\tilde{b} \ge \tilde{c})}} - v_{{p(\tilde{b} \ge \tilde{c})}} > 0\) hold with the third and the second case in Eq. (3), respectively, i.e., \(a^{ - } \le b^{ - } \le b^{ + } \le a^{ + }\) and \(c^{ - } \le b^{ - } \le c^{ + } \le b^{ + }\), so \(c^{ + } \le a^{ + }\), \(a^{ + } - b^{ + } + a^{ - } - b^{ - } > 0\) and \(b^{ + } - c^{ + } + b^{ - } - c^{ - } > 0\).

If a − ≤ c −, then \((a^{ + } - c^{ + } + a^{ - } - c^{ - } ) - (b^{ + } - c^{ + } + b^{ - } - c^{ - } ) = a^{ + } - b^{ + } + a^{ - } - b^{ - } > 0\), thus \(\mu_{{p(\tilde{a} \ge \tilde{c})}} - v_{{p(\tilde{a} \ge \tilde{c})}} > 0\), i.e., \(\tilde{a} >_{{\mu_{{p(\tilde{a} \ge \tilde{c})}} }} \tilde{c}\);

If c − ≤ a − ≤ c +, because \(c^{ + } \le a^{ + }\), the conclusion that \(a^{ + } - c^{ + } + a^{ - } - c^{ - } > 0\) is direct, thus \(\mu_{{p(\tilde{a} \ge \tilde{c})}} - v_{{p(\tilde{a} \ge \tilde{c})}} > 0\), i.e., \(\tilde{a} >_{{\mu_{{p(\tilde{a} \ge \tilde{c})}} }} \tilde{c}\);

Case 9

If both \(\mu_{{p(\tilde{a} \ge \tilde{b})}} - v_{{p(\tilde{a} \ge \tilde{b})}} > 0\) and \(\mu_{{p(\tilde{b} \ge \tilde{c})}} - v_{{p(\tilde{b} \ge \tilde{c})}} > 0\) hold with the third case in Eq. (4), i.e., \(a^{ - } \le b^{ - } \le b^{ + } \le a^{ + }\) and \(b^{ - } \le c^{ - } \le c^{ + } \le b^{ + }\), so \(a^{ - } \le b^{ - } \le c^{ - } \le c^{ + } \le b^{ + } \le a^{ + }\) and \(a^{ + } - b^{ + } + a^{ - } - b^{ - } > 0\), \(b^{ + } - c^{ + } + b^{ - } - c^{ - } > 0\), the result that \(a^{ + } - c^{ + } + a^{ - } - c^{ - } > 0\) is obvious, thus \(\mu_{{p(\tilde{a} \ge \tilde{c})}} - v_{{p(\tilde{a} \ge \tilde{c})}} > 0\), i.e., \(\tilde{a} >_{{\mu_{{p(\tilde{a} \ge \tilde{c})}} }} \tilde{c}\).

According to the cases mentioned above, corresponding to different situations, the transitivity of the proposed intuitionistic fuzzy possibility degree holds, which proves Theorem 1. □

Next, we can obtain the following properties of intuitionistic fuzzy comparing law for interval numbers.

Theorem 2

The intuitionistic fuzzy comparing law for interval numbers defined by Eq. (3) is a legal partial order.

Proof

We just verify the properties of reflexivity, antisymmetry, and transitivity.

  1. (1)

    (Reflexivity): Let Φ be the set of all interval numbers, according to Eq. (3), we have \(\forall \;\tilde{a} \in \Phi ,\;p\left( {\tilde{a} \ge \tilde{a}} \right) = \left\langle {0,\;0,\;1} \right\rangle\), thus the conclusion that \(\tilde{a} \ge \tilde{a}\) holds.

  2. (2)

    (Antisymmetry): For any \(\tilde{a},\;\tilde{b} \in \Phi\), if \(\tilde{a} \ge \tilde{b}\) and \(\tilde{b} \ge \tilde{a}\), then we have \(\mu_{{p\left( {\tilde{a} \ge \tilde{b}} \right)}} - v_{{p\left( {\tilde{a} \ge \tilde{b}} \right)}} \ge 0\) and \(\mu_{{p\left( {\tilde{b} \ge \tilde{a}} \right)}} - v_{{p\left( {\tilde{b} \ge \tilde{a}} \right)}} \ge 0\), since \(\mu_{{p\left( {\tilde{a} \ge \tilde{b}} \right)}} = v_{{p\left( {\tilde{b} \ge \tilde{a}} \right)}}\) and \(v_{{p\left( {\tilde{a} \ge \tilde{b}} \right)}} = \mu_{{p\left( {\tilde{b} \ge \tilde{a}} \right)}}\), thus \(\mu_{{p\left( {\tilde{b} \ge \tilde{a}} \right)}} = v_{{p\left( {\tilde{b} \ge \tilde{a}} \right)}}\), i.e., \(\tilde{a}\sim \tilde{b}\).

  3. (3)

    (Transitivity) According to Theorem 1, this property is obvious.

Thus, the rationality of intuitionistic fuzzy partial order for interval numbers can be verified. \(\square\)

Theorem 3

Let \(\tilde{a} = [a^{ - } ,\;a^{ + } ]\), \(\tilde{b} = [b^{ - } ,\;\;b^{ + } ]\) and \(\tilde{c} = [c^{ - } ,\;\;c^{ + } ]\) be any three positive interval numbers, and \(k > 0\), then the following results hold.

  1. (1)

    \(p(k\tilde{a} \ge k\tilde{b}) = p(\tilde{a} \ge \tilde{b})\);

  2. (2)

    If \(\tilde{a} >_{{\mu_{{p(\tilde{a} \ge \tilde{b})}} }} \tilde{b}\), then \(\tilde{a} + \tilde{c} >_{{\mu_{{p(\tilde{a} + \tilde{c} \ge \tilde{b} + \tilde{c})}} }} \tilde{b} + \tilde{c}\). Especially, if \(\tilde{c}\) degenerates to be a scalar, then \(\tilde{a} + c >_{{\mu_{{p(\tilde{a} + c \ge \tilde{b} + c)}} }} \tilde{b} + c\).

Proof

  1. (1)

    Let \(\tilde{d} = k\tilde{a}\), \(\tilde{e} = k\tilde{b}\), and \(\tilde{d} = [d^{ - } ,\;\;d^{ + } ]\), \(\tilde{e} = [e^{ - } ,\;\;e^{ + } ]\), then

$$d^{ - } = ka^{ - } ,d^{ + } = ka^{ + } ,e^{ - } = kb^{ - } ,e^{ + } = kb^{ + } .$$

There are three cases:

Case 1

If a − ≥ b +, since \(k > 0\), then \(ka^{ - } \ge kb^{ + }\), i.e., \(e^{ - } \ge d^{ + }\). According to Eq. (3), we get that

$$\mu _{{p(\tilde{a} \ge \tilde{b})}} = 1,v_{{p(\tilde{a} \ge \tilde{b})}} = 0,\pi _{{p(\tilde{a} \ge \tilde{b})}} = 0,\,{\text{and}}\,\mu _{{p(\tilde{d} \ge \tilde{e})}} = 1,v_{{p(\tilde{d} \ge \tilde{e})}} = 0,\pi _{{p(\tilde{d} \ge \tilde{e})}} = 0.$$

Thus, \(p(\tilde{d} \ge \tilde{e}) = p(\tilde{a} \ge \tilde{b})\), i.e., \(p(k\tilde{a} \ge k\tilde{b}) = p(\tilde{a} \ge \tilde{b})\).

Case 2

If \(b^{ - } \le a^{ - } \le b^{ + } \le a^{ + }\), since \(k > 0\), then \(kb^{ - } \le ka^{ - } \le kb^{ + } \le ka^{ + }\), i.e., \(e^{ - } \le d^{ - } \le e^{ + } \le d^{ + }\).

According to Eq. (3), we get that

$$\mu _{{p(\tilde{d},\tilde{e})}} = \frac{{d^{ + } - e^{ + } + d^{ - } - e^{ - } }}{{d^{ + } - e^{ - } }} = \frac{{ka^{ + } - kb^{ + } + ka^{ - } - kb^{ - } }}{{ka^{ + } - kb^{ - } }} = \frac{{a^{ + } - b^{ + } + a^{ - } - b^{ - } }}{{a^{ + } - b^{ - } }} = \mu _{{p(\tilde{a},\tilde{b})}} .$$

Similarly, \(v_{{p(\tilde{d} \ge \tilde{e})}} = v_{{p(\tilde{a} \ge \tilde{b})}} ,\pi _{{p(\tilde{d} \ge \tilde{e})}} = \pi _{{p(\tilde{a} \ge \tilde{b})}} .\). Thus,

$$p(\tilde{d} \ge \tilde{e}) = p(\tilde{a} \ge \tilde{b}),i.e., p(k\tilde{a} \ge k\tilde{b}) = p(\tilde{a} \ge \tilde{b}).$$

Case 3

If \(a^{ - } \le b^{ - } \le b^{ + } \le a^{ + }\), since \(k > 0\), then \(ka^{ - } \le kb^{ - } \le kb^{ + } \le ka^{ + }\), i.e., \(d^{ - } \le e^{ - } \le e^{ + } \le d^{ + }\).

According to Eq. (3), then

$$\mu _{{p(\tilde{d} \ge \tilde{e})}} = \frac{{d^{ + } - e^{ + } }}{{d^{ + } - d^{ - } }} = \frac{{ka^{ + } - kb^{ + } }}{{ka^{ + } - ka^{ - } }} = \frac{{a^{ + } - b^{ + } }}{{a^{ + } - a^{ - } }} = \mu _{{p(\tilde{a} \ge \tilde{b})}} .$$

In similar way, we have \(v_{{p(\tilde{d} \ge \tilde{e})}} = v_{{p(\tilde{a} \ge \tilde{b})}} ,\pi _{{p(\tilde{d} \ge \tilde{e})}} = \pi _{{p(\tilde{a} \ge \tilde{b})}} .\)

Thus, \(p(\tilde{d} \ge \tilde{e}) = p(\tilde{a} \ge \tilde{b}){\rm ,\,i.e.,} p(k\tilde{a} \ge k\tilde{b}) = p(\tilde{a} \ge \tilde{b})\), which completes the proof. □

  1. (2)

    According to the additive operation for interval numbers,

$$\tilde{a} + \tilde{c} = [a^{ - } + c^{ - } ,a^{ + } + c^{ + } ],\tilde{b} + \tilde{c} = [b^{ - } + c^{ - } ,b^{ + } + c^{ + } ].$$

Herein, three situations are taken into account:

  1. (1)

    If \(b^{ - } \le b^{ + } \le a^{ - } \le a^{ + }\), i.e., \(\mu_{{p\left( {\tilde{a} \ge \tilde{b}} \right)}} = 1\), then

If \(a^{ - } + c^{ - } \ge b^{ + } + c^{ + }\), thus \(\mu_{{p\left( {\tilde{a} + \tilde{c} \ge \tilde{b} + \tilde{c}} \right)}} = 1\), i.e., \(\tilde{a} + \tilde{c} >_{{\mu_{{p(\tilde{a} + \tilde{c} \ge \tilde{b} + \tilde{c})}} }} \tilde{b} + \tilde{c}\); If \(a^{ - } + c^{ - } < b^{ + } + c^{ + }\), then there would be an intersection between \(\tilde{a} + \tilde{c}\) and \(\tilde{b} + \tilde{c}\), for the reason that

$$[a^{ + } + c^{ + } - (b^{ + } + c^{ + } )] + [a^{ - } + c^{ - } - (b^{ - } + c^{ - } )] = a^{ + } - b^{ + } + a^{ - } - b^{ - } > 0,$$

Thus, \(\mu_{{p(\tilde{a} + \tilde{c} \ge \tilde{b} + \tilde{c})}} - v_{{p(\tilde{a} + \tilde{c} \ge \tilde{b} + \tilde{c})}} > 0\), i.e., \(\tilde{a} + \tilde{c} > \mu _{{p(\tilde{a} + \tilde{c} \ge \tilde{b} + \tilde{c})}} \tilde{b} + \tilde{c}.\)

  1. (2)

    If \(b^{ - } \le a^{ - } \le b^{ + } \le a^{ + } ,\) then we have \(b^{ - } + c^{ - } \le a^{ - } + c^{ - } \le b^{ + } + c^{ + } \le a^{ + } + c^{ + } ,\) because

$$[a^{ + } + c^{ + } - (b^{ + } + c^{ + } )] + [a^{ - } + c^{ - } - (b^{ - } + c^{ - } )] = a^{ + } - b^{ + } + a^{ - } - b^{ - } > 0,$$

So, \(\mu_{{p(\tilde{a} + \tilde{c} \ge \tilde{b} + \tilde{c})}} - v_{{p(\tilde{a} + \tilde{c} \ge \tilde{b} + \tilde{c})}} > 0\), i.e., \(\tilde{a} + \tilde{c} >_{{\mu_{{p(\tilde{a} + \tilde{c} \ge \tilde{b} + \tilde{c})}} }} \tilde{b} + \tilde{c}\).

  1. (3)

    If \(a^{ - } \le b^{ - } \le b^{ + } \le a^{ + }\) and \(\mu_{{p\left( {\tilde{a} \ge \tilde{b}} \right)}} - v_{{p\left( {\tilde{a} \ge \tilde{b}} \right)}} > 0\), then \(a^{ + } - b^{ + } + a^{ - } - b^{ - } > 0\) and \(a^{ - } + c^{ - } \le b^{ - } + c^{ - } \le b^{ + } + c^{ + } \le a^{ + } + c^{ + }\), since

$$[a^{ + } + c^{ + } - (b^{ + } + c^{ + } )] + [a^{ - } + c^{ - } - (b^{ - } + c^{ - } )] = a^{ + } - b^{ + } + a^{ - } - b^{ - } > 0,$$

So, \(\mu _{{p(\tilde{a} + \tilde{c} \ge \tilde{b} + \tilde{c})}} - v_{{p(\tilde{a} + \tilde{c} \ge \tilde{b} + \tilde{c})}} > 0,\) i.e., \(\tilde{a} + \tilde{c} >_{{\mu_{{p(\tilde{a} + \tilde{c} \ge \tilde{b} + \tilde{c})}} }} \tilde{b} + \tilde{c}\).

The situation that \(\tilde{a}\) and \(\tilde{b}\) exchange their locations could be proofed similarly. Therefore, the conclusion that \(\tilde{a} + \tilde{c} >_{{\mu_{{p(\tilde{a} + \tilde{c} \ge \tilde{b} + \tilde{c})}} }} \tilde{b} + \tilde{c}\) holds.

Especially, if \(\tilde{c}\) degenerates to be a scalar, which is a special case of \(\tilde{c}\), so the result is also correct, which proves the theorem. \(\square\)

To illustrate the application of the proposed intuitionistic fuzzy preference relation, the following example is considered:

Example 1

Let us take a simple example of choosing the best minimum from among the cost intervals (Segupta and Pal [30]): A = [20, 40], B = [22, 32], C = [26, 34].

According to Eq. (3), the comparing results can be listed as follows:

$$p(A \ge B) = \left\langle {0.4,0.1,0.5} \right\rangle ,p(A \ge C) = \left\langle {0.3,0.3,0.4} \right\rangle ,p(B \ge C) = \left\langle {0,0.33,0.67} \right\rangle .$$

According to Definition 5, it can be obtained that 

$$A >_{0.4} B,\,C >_{0.33}\,B \quad {\rm and } \quad A\sim_{0.4} C.$$

Meanwhile, it can be seen that \(\mu _{{p(\tilde{c},\tilde{b})}} - v_{{p(\tilde{c},\tilde{b})}} = 0.33\) and \(\mu _{{p(\tilde{a},\tilde{b})}} - v_{{p(\tilde{a},\tilde{b})}} = 0.30\), i.e., C is dominated by B with greater degree.

Therefore, \(C\) is the best choice, which is the same as in Segupta and Pal [30].

Example 2

Assume that \(\tilde{a}_{1}\) = [1, 4], \(\tilde{a}_{2}\) = [2, 4], \(\tilde{a}_{3}\) = [1, 5], \(\tilde{a}_{4}\)=[0, 6] are all interval numbers, then

$$p(\tilde{a}_{1} \ge \tilde{a}_{2} ) = \left\langle {0,\frac{1}{3},\frac{2}{3}} \right\rangle,\,p(\tilde{a}_{3} \ge \tilde{a}_{1} ) = \left\langle {\frac{1}{4},0,\frac{3}{4}} \right\rangle,\,p(\tilde{a}_{4} \ge \tilde{a}_{1} ) = \left\langle {\frac{1}{3},\frac{1}{6},\frac{1}{2}} \right\rangle,\,p(\tilde{a}_{3} \ge \tilde{a}_{2} ) = \left\langle {\frac{1}{4},\frac{1}{4},\frac{1}{2}} \right\rangle,\,p(\tilde{a}_{4} \ge \tilde{a}_{2} ) = \left\langle {\frac{1}{3},\frac{1}{3},\frac{1}{3}} \right\rangle,\,p(\tilde{a}_{4} \ge \tilde{a}_{3} ) = \left\langle {\frac{1}{6},\frac{1}{6},\frac{2}{3}} \right\rangle.$$

According to Definition 5, we can conclude that \(\tilde{a}_{2} \sim_{0.5} \tilde{a}_{3}\), \(\tilde{a}_{2} \sim_{0.333} \tilde{a}_{4}\), \(\tilde{a}_{3} \sim_{0.667} \tilde{a}_{4}\), \(\tilde{a}_{2} >_{0.333} \tilde{a}_{1}\), \(\tilde{a}_{3} >_{0.25} \tilde{a}_{1},\) and \(\tilde{a}_{4} >_{0.333} \tilde{a}_{1}\).

As a result, three intervals \(\tilde{a}_{2}\), \(\tilde{a}_{3},\) and \(\tilde{a}_{4}\) are equivalent to each other at different levels, and \(\tilde{a}_{2}\), \(\tilde{a}_{3},\) and \(\tilde{a}_{4}\) are all greater than \(\tilde{a}_{1}\).

When coming to the ranking of \(\tilde{a}_{2}\), \(\tilde{a}_{3},\) and \(\tilde{a}_{4}\), a decision maker needs more information, such as the attitude of the decision maker to the risk (i.e., a risk lover may choose the interval with longer width, while a risk averter may select the inverse value).

4 The Application of Proposed IFPD Method to FMADM

In this section, we introduce a novel method for fuzzy MADM ([43, 55, 57, 58]) on the basis of proposed intuitionistic fuzzy possibility degree of interval-valued fuzzy numbers.

Let U = {u 1, u 2, \(\cdots\), u m } is the set of alternatives, A = {a 1, a 2, \(\cdots\), a n } is the attribute set with corresponding weight vector W = (w 1, w 2, \(\cdots\), w n )T, and the matrix \({\tilde{\mathbf{V}}} = (\tilde{v}_{ij} )_{m \times n} = ([v_{ij}^{ - } ,\;\;v_{ij}^{ + } ])_{m \times n}\) is the attribute values of the i-th alternative with respect to the j-th attribute, where the elements in \(\tilde{\varvec{V}}\) are interval-valued fuzzy numbers, i = 1, 2, \(\cdots\), m, j = 1, 2, \(\cdots\), n.

To select the best alternative(s), we propose the following decision procedure:

  • Step 1 Constructing the normalized decision matrix \(\tilde{\hat{\varvec{V}}}\) of \(\tilde{\varvec{V}}\), where \(\tilde{\hat{\varvec{V}}} = (\tilde{\hat{v}}_{ij} )_{m \times n}\) and \(\tilde{\hat{v}}_{ij} = ([\hat{v}_{ij}^{ - } ,\;\;\hat{v}_{ij}^{ + } ])_{m \times n}\).

    Since the types of attributes are not the same, thus we need to standardize the attribute values so that the affection of physical dimension can be eliminated.

    According to the type of each attribute, the normalization can be finished by

    $$\tilde{\hat{v}}_{ij} = \left[ {{{v_{ij}^{ - } } \mathord{\left/ {\vphantom {{v_{ij}^{ - } } {\sum\limits_{i = 1}^{n} {v_{ij}^{ + } } }}} \right. \kern-0pt} {\sum\limits_{i = 1}^{n} {v_{ij}^{ + } } }}\;\;\;{{v_{ij}^{ + } } \mathord{\left/ {\vphantom {{v_{ij}^{ + } } {\sum\limits_{i = 1}^{n} {v_{ij}^{ - } } }}} \right. \kern-0pt} {\sum\limits_{i = 1}^{n} {v_{ij}^{ - } } }}} \right], {\text{if}}\, {a_{j}}\, {\text{is a benefit type attribute,}}$$
    (9)

    or

    $$\tilde{\hat{v}}_{ij} = \left[ {{{\frac{1}{{v_{ij}^{ + } }}} \mathord{\left/ {\vphantom {{\frac{1}{{v_{ij}^{ + } }}} {\sum\limits_{i = 1}^{n} {\frac{1}{{v_{ij}^{ - } }}} }}} \right. \kern-0pt} {\sum\limits_{i = 1}^{n} {\frac{1}{{v_{ij}^{ - } }}} }}\;\;\;{{\frac{1}{{v_{ij}^{ - } }}} \mathord{\left/ {\vphantom {{\frac{1}{{v_{ij}^{ - } }}} {\sum\limits_{i = 1}^{n} {\frac{1}{{v_{ij}^{ + } }}} }}} \right. \kern-0pt} {\sum\limits_{i = 1}^{n} {\frac{1}{{v_{ij}^{ + } }}} }}} \right],\, {\text{if}}\,a_{j}\,{\text{is a cost type attribute,}}$$
    (10)

    where i = 1, 2, \(\cdots\), m, j = 1, 2, \(\cdots\), n.

  • Step 2 Calculating the weighting vector of attributes based on the proposed IFPD method.

Suppose that \(\tilde{\hat{v}}_{j} = \left( {\tilde{\hat{v}}_{ij} } \right)_{m \times 1} ,\;j = 1,\;2,\; \cdots ,\;n\) represents the decision information of all alternatives with respect to the j-th attribute. We can derive the comparisons among all alternatives according to \(\tilde{\hat{v}}_{j}\), which can be given by

$$\begin{aligned}&\qquad\qquad \begin{array}{*{20}l}\quad\quad {u_{1} } &\quad\quad\quad\quad\quad {u_{2} } &\quad\,\, {\cdots } & \quad\quad {u_{m} } \\ \end{array} \hfill \\ P\left( {\tilde{\hat{v}}_{j} } \right) &= \begin{array}{*{20}l} {u_{1} } \\ {u_{2} } \\ \vdots \\ {u_{m} } \\ \end{array} \left( {\begin{array}{*{20}c} {\left\langle {0,\;0,\;1} \right\rangle } & {p(\tilde{\hat{v}}_{ 1j} \ge \tilde{\hat{v}}_{ 2j} )} & \cdots & {p(\tilde{\hat{v}}_{ 1j} \ge \tilde{\hat{v}}_{{{\text{m}}j}} )} \\ {p(\tilde{\hat{v}}_{ 2j} \ge \tilde{\hat{v}}_{ 1j} )} & {\left\langle {0,\;0,\;1} \right\rangle } & \cdots & {p(\tilde{\hat{v}}_{2j} \ge \tilde{\hat{v}}_{{{\text{m}}j}} )} \\ \cdots & \cdots & \cdots & \cdots \\ {p(\tilde{\hat{v}}_{{{\text{m}}j}} \ge \tilde{\hat{v}}_{ 1j} )} & {p(\tilde{\hat{v}}_{{{\text{m}}j}} \ge \tilde{\hat{v}}_{2j} )} & \cdots & {\left\langle {0,\;0,\;1} \right\rangle } \\ \end{array} } \right),\;j = 1,\;2,\; \cdots ,\;n \hfill \\ \end{aligned} ,$$
(11)

and denoted as \(P\left( {\tilde{\hat{v}}_{j} } \right) = (p_{kl}^{j} )_{m \times m} = (p(\tilde{r}_{kj} \ge \tilde{r}_{lj} ))_{m \times m}\).

It’s worth noting that \(P\left( {\tilde{\hat{v}}_{j} } \right),\;j = 1,\;2,\; \cdots ,\;n\) is a symmetrical intuitionistic fuzzy preference relation. Correspondingly, the FMADM with IVFNs transforms to be a group decision making with intuitionistic fuzzy preference relations (Wu and Chiclana [38]; Behret [4]).

To determine the weighting vector of all attributes, by using Definition 3 and Eq. (11), we first calculate the total distance of the j-th attribute with respect to the other attributes according to

$$d_{j} = \sum\limits_{\begin{subarray}{l} q = 1 \\ q \ne j \end{subarray} }^{n} {\left( {\frac{1}{m(m - 1)}\sum\limits_{k = 1}^{m} {\sum\limits_{\begin{subarray} {l} l = 1 \\ l \ne k \end{subarray} }^{m} {d(p_{kl}^{j} ,p_{kl}^{q} )} } } \right)} ,$$
(12)

where \(\frac{1}{m(m - 1)}\sum\nolimits_{k = 1}^{m} {\sum\limits_{\begin{subarray}{l} l = 1 \\ l \ne k \end{subarray} }^{m} {d(p_{kl}^{j} ,p_{kl}^{q} )} }\) represents the average distance of the elements under the j-th and the q-th attributes. It is obvious that Eq. (12) can be rewritten as

$$d_{j} = \frac{1}{m(m - 1)}\sum\limits_{\begin{subarray}{l} q = 1 \\ q \ne j \end{subarray} }^{n} {\sum\limits_{k = 1}^{m} {\sum\limits_{\begin{subarray}{l} l = 1 \\ l \ne k \end{subarray} }^{m} {d(p_{kl}^{j} ,p_{kl}^{q} )} } } .$$
(13)

Noting that without any prior information, the less of the value \(d_{j}\) is, the less of the weight \(w_{j}\) would be, because the decision information under \(a_{j}\) would also be useless in the decision process. In order to obtain a final decision, we get the weights of \(a_{j} ,\;j = 1,\;2,\; \cdots ,\;n\) by

$$w_{j} = {{d_{j} } \mathord{\left/ {\vphantom {{d_{j} } {\sum\limits_{j = 1}^{n} {d_{j} } }}} \right. \kern-0pt} {\sum\limits_{j = 1}^{n} {d_{j} } }} .$$
(14)

  • Step 3 Aggregating the decision information for all alternatives using the weights obtained in Step 2.

We utilize the weighted average mean of IVFNs to realize the aggregation process, i.e.,

$$\tilde{r}_{i} = \sum\limits_{j = 1}^{n} {w_{j} \tilde{\hat{v}}_{ij} } ,\;i = 1,\;2,\; \cdots ,\;m .$$
(15)

  • Step 4 Comparing the final decision information using the proposed intuitionistic fuzzy preference relation.

Similar to the preference relation matrix obtained in Step 2, we get corresponding preference relation of the final decision information according to

$$\begin{aligned} &\quad\quad\quad \begin{array}{*{20}l} \quad\quad\quad {u_{1} } &\quad\quad\quad\quad {u_{2} } &\quad\,\,\,\,\, {\cdots } & {\;\;\;\;\;\;u_{m} } \\ \end{array} \hfill \\ P &= \begin{array}{*{20}c} {u_{1} } \\ {u_{2} } \\ \vdots \\ {u_{m} } \\ \end{array} \left( {\begin{array}{*{20}c} {\left\langle {0,\;0,\;1} \right\rangle } & {p(\tilde{r}_{1} \ge \tilde{r}_{2} )} & \cdots & {p(\tilde{r}_{1} \ge \tilde{r}_{m} )} \\ {p(\tilde{r}_{2} \ge \tilde{r}_{1} )} & {\left\langle {0,\;0,\;1} \right\rangle } & \cdots & {p(\tilde{r}_{2} \ge \tilde{r}_{m} )} \\ \cdots & \cdots & \cdots & \cdots \\ {p(\tilde{r}_{m} \ge \tilde{r}_{1} )} & {p(\tilde{r}_{m} \ge \tilde{r}_{2} )} & \cdots & {\left\langle {0,\;0,\;1} \right\rangle } \\ \end{array} } \right) \hfill \\ \end{aligned} .$$
(16)

To avoid comparing the intuitionistic fuzzy values one by one, we utilize the optimization model proposed by Behret [4] to derive the ranking values of alternatives, which can be shown as follows:

$$\begin{aligned} \;\;\;\;\;f(\lambda_{1} ,\;\lambda_{2} ,\; \cdots ,\;\lambda_{m} ) = {\rm min}\sum\limits_{i = 1}^{m} {\sum\limits_{j = i + 1}^{m} {\left( {d_{ij}^{ - } + d_{ij}^{ + } } \right)} } \hfill \\ \;\;\;\;s.t.\; \left\{ \begin{aligned} 0.5(\lambda_{i} - \lambda_{j} + 1) + d_{ij}^{ - } \ge \mu_{{p(\tilde{r}_{i} \ge \tilde{r}_{j} )}}; \hfill \\ 0.5(\lambda_{i} - \lambda_{j} + 1) - d_{ij}^{ + } \le 1 - v_{{p(\tilde{r}_{i} \ge \tilde{r}_{j} )}}; \hfill \\ \sum\limits_{i = 1}^{m} {\lambda_{i} } = 1,\;\lambda_{i} \ge 0,\;i = 1,\;2,\; \cdots ,m; \hfill \\ d_{ij}^{ - } ,\;d_{ij}^{ + } \ge 0,\;i = 1,\;2,\; \cdots ,\;m,\;j = i + 1,\; \cdots ,\;m. \hfill \\ \end{aligned} \right. \hfill \\ \end{aligned}$$
(17)

where \(\lambda_{i}\) represents the weight and the ranking value of \(u_{i} ,\;i = 1,\;2,\; \cdots ,\;m\).

Remark 1

Eq. (17) is suitable to the case that there are many alternatives need to be compared. However, if the number of alternatives is very limited, we can give the order of alternatives directly by using the proposed intuitionistic fuzzy values.

  • Step 5 Ranking \(\lambda_{i} \left( {i = 1,2, \cdots ,m} \right)\) in descending order and selecting the best alternative.

  • Step 6 End

According to the proposed decision procedure, in the stage of obtaining weighting vector, by using the developed intuitionistic fuzzy order, the differences among different attributes are presented by the internal compassions of alternatives. The advantage of the weighting method is that the obtained weighting vector can show the differences from the microcosmic comparisons and the holistic perspective. By analyzing the decision algorithm, the complexity is \(O(N^{3} )\), \(N = \hbox{max} \left\{ {m,\;n} \right\}\), where \(m,\;n\) are the number of alternatives and attributes, respectively.

Recently, Xu and Liao [61] gave a survey on approaches to decision making with intuitionistic fuzzy preference relations, so the proposed method provides a perspective for solving decision making with interval fuzzy numbers. As a direct extension, one may consider an interval-valued intuitionistic fuzzy order for interval-valued hesitant fuzzy sets so that the interval-valued intuitionistic fuzzy preference relations (Liao et al. 62]) can be utilized.

5 Case Study

5.1 Backgrounds

Maintainability is of great concern in the products’ research and development process, which should be considered from multiple indicators, such as the total structure of the system, the configuration and connection for each part, standardization and modularization, etc. As a result, the user could recover all the functions of the product in time when it meets some faults. (Adapted from Xu et al. [39])

Laptop has been a common and useful tool in daily life. Its main functions are small in size, light weight, and convenient to carry. In order to make sure that the bought laptop can work normally and efficiently, the customers have to consider several maintainability designs for their alternatives. Some indices are designed in the evaluation of laptop’s maintainability design, life cycle cost (unit: dollars) (a 1), life expectancy (unit: hundred hours) (a 2), mean time to repair (unit: hours) (a 3), availability (a 4), and comprehensive performance (a 5).

Herein, our laboratory is preparing to purchase a certain amount of laptops; after inspecting, three brands U = {u 1, u 2, u 3} are selected for future consideration. Their prices are of the same level, but their maintainability designs are not the same. Table 1 shows the evaluation interval values for each manufacture with respect to the five indices mentioned above.

Table 1 Evaluation interval values for each alternative with respect to indices
Full size table

Among them, the life cycle cost and the mean time to repair are both cost types, while the others are all benefit types. We are now making the right decision for the laboratory using our mentioned approach.

5.2 Decision Process and Results

The following processes are presented to illustrate the proposed multi-attribute decision-making method based on intuitionistic fuzzy possibility degree of interval numbers.

  • Step 1 Constructing the normalized decision matrix based on Eqs. (9), (10).

According to the types of attributes, we can convert decision matrix \(\tilde{V}\) into standardized decision matrix \(\tilde{\hat{V}}\), which is shown in Table 2.

Table 2 Standardized decision matrix \(\tilde{\hat{V}}\)
Full size table

  • Step 2 Calculating the weighting vector of attributes based on the proposed IFPD method.

According to Eq. (11), we have

$$\begin{aligned} &\qquad\qquad\qquad\begin{array}{*{20}l} \quad{u_{1} } & \quad\quad\quad\quad\quad\quad\quad{u_{2} } & \quad\quad\quad\quad\quad\quad\quad\quad{u_{3} } \\ \end{array} \hfill \\ P\left( {\tilde{\hat{v}}_{1} } \right) &= \begin{array}{*{20}c} {u_{1} } \\ {u_{2} } \\ {u_{3} } \\ \end{array} \left( {\begin{array}{*{20}c} {\left\langle {0,\;0,\;1} \right\rangle } & {\left\langle {0.9067,\;0,\;0.0933} \right\rangle } & {\left\langle {0,\;1,\;0} \right\rangle } \\ {\left\langle {0,\;0.9067,\;0.0933} \right\rangle } & {\left\langle {0,\;0,\;1} \right\rangle } & {\left\langle {0,\;0.7292,\;0.2708} \right\rangle } \\ {\left\langle {1,\;0,\;0} \right\rangle } & {\left\langle {0.7292,\;0,\;0.2708} \right\rangle } & {\left\langle {0,\;0,\;1} \right\rangle } \\ \end{array} } \right); \hfill \\ \end{aligned}$$
$$\begin{aligned} &\begin{array}{*{20}l} \quad\quad\quad\quad\quad{u_{1} } & \quad\quad\quad{u_{2} } & \quad\quad{u_{3} } \\ \end{array} \hfill \\ P\left( {\tilde{\hat{v}}_{2} } \right) &= \begin{array}{*{20}c} {u_{1} } \\ {u_{2} } \\ {u_{3} } \\ \end{array} \left( {\begin{array}{*{20}c} {\left\langle {0,\;0,\;1} \right\rangle } & {\left\langle {0,\;1,\;0} \right\rangle } & {\left\langle {0,\;1,\;0} \right\rangle } \\ {\left\langle {1,\;0,\;0} \right\rangle } & {\left\langle {0,\;0,\;1} \right\rangle } & {\left\langle {1,\;0,\;0} \right\rangle } \\ {\left\langle {1,\;0,\;0} \right\rangle } & {\left\langle {0,\;1,\;0} \right\rangle } & {\left\langle {0,\;0,\;1} \right\rangle } \\ \end{array} } \right); \hfill \\ \end{aligned}$$
$$\begin{aligned} &\begin{array}{*{20}l} \quad\quad\quad\quad\quad\quad\quad{u_{1} } & \quad\quad\quad\quad\quad{u_{2} } & \quad\quad\quad\quad\quad{u_{3} } \\ \end{array} \hfill \\ P\left( {\tilde{\hat{v}}_{3} } \right) &= \begin{array}{*{20}c} {u_{1} } \\ {u_{2} } \\ {u_{3} } \\ \end{array} \left( {\begin{array}{*{20}c} {\left\langle {0,\;0,\;1} \right\rangle } & {\left\langle {1,\;0,\;0} \right\rangle } & {\left\langle {0.4206,\;0,\;0.5794} \right\rangle } \\ {\left\langle {0,\;1,\;0} \right\rangle } & {\left\langle {0,\;0,\;1} \right\rangle } & {\left\langle {0,\;1,\;0} \right\rangle } \\ {\left\langle {0,\;0.4206,\;0.5794} \right\rangle } & {\left\langle {1,\;0,\;0} \right\rangle } & {\left\langle {0,\;0,\;1} \right\rangle } \\ \end{array} } \right); \hfill \\ \end{aligned}$$
$$\begin{aligned} &\begin{array}{*{20}l} \quad\quad\quad\quad\quad\quad{u_{1} } & \quad\quad\quad\quad\quad\quad\quad\quad{u_{2} } &\quad\quad\quad\quad\quad\quad\quad {u_{3} } \\ \end{array} \hfill \\ P\left( {\tilde{\hat{v}}_{4} } \right) &= \begin{array}{*{20}c} {u_{1} } \\ {u_{2} } \\ {u_{3} } \\ \end{array} \left( {\begin{array}{*{20}c} {\left\langle {0,\;0,\;1} \right\rangle } & {\left\langle {0,\;0.3158,\;0.6842} \right\rangle } & {\left\langle {0,\;0.96,\;0.04} \right\rangle } \\ {\left\langle {0.3158,\;0,\;0.6842} \right\rangle } & {\left\langle {0,\;0,\;1} \right\rangle } & {\left\langle {0,\;0.72,\;0.28} \right\rangle } \\ {\left\langle {0.96,\;0,\;0.04} \right\rangle } & {\left\langle {0.72,\;0,\;0.28} \right\rangle } & {\left\langle {0,\;0,\;1} \right\rangle } \\ \end{array} } \right); \hfill \\ \end{aligned}$$
$$\begin{aligned} &\begin{array}{*{20}l} \quad\quad\quad\quad\quad\quad\quad{u_{1} } & \quad\quad\quad\quad\quad\quad\quad\quad{u_{2} } & \quad\quad\quad\quad\quad\quad\quad{u_{3} } \\ \end{array} \hfill \\ P\left( {\tilde{\hat{v}}_{5} } \right) &= \begin{array}{*{20}c} {u_{1} } \\ {u_{2} } \\ {u_{3} } \\ \end{array} \left( {\begin{array}{*{20}c} {\left\langle {0,\;0,\;1} \right\rangle } & {\left\langle {0,\;0.8261,\;0.1739} \right\rangle } & {\left\langle {0,\;1,\;0} \right\rangle } \\ {\left\langle {0.8261,\;0,\;0.1739} \right\rangle } & {\left\langle {0,\;0,\;1} \right\rangle } & {\left\langle {0,\;0.9661,\;0.0339} \right\rangle } \\ {\left\langle {1,\;0,\;0} \right\rangle } & {\left\langle {0.9661,\;0,\;0.0339} \right\rangle } & {\left\langle {0,\;0,\;1} \right\rangle } \\ \end{array} } \right). \hfill \\ \end{aligned}$$

Then, according to Eq. (13), the total distance of the j-th (j = 1, 2, \(\cdots\), 5) attribute with respect to the other attributes can be calculated. The results are shown in the following,

$$d_{ 1} = 3. 6 4 2 4,d_{ 2} = 5. 2 6 5 4,d_{ 3} = 5. 7 5 8 7,d_{ 4} = 3. 8 1 10,d_{ 5} = 4.00 2 8.$$

Thus, the weighting vector of the attributes can computed according to Eq. (14), we have

$$w_{ 1} = 0. 1 6 20,w_{ 2} = 0. 2 3 4 2,w_{ 3} = 0. 2 5 6 2,w_{ 4} = 0. 1 6 9 5,w_{ 5} = 0. 1 7 8 1.$$

  • Step 3 Aggregating the decision information for all alternatives using the weights obtained in Step 2.

By using Eq. (15) and the operational laws of IVFNs, the final aggregation interval values for alternatives are,

$$\tilde{r}_{1} =[0.5366, 0.6013],\,\tilde{r}_{2} =[0.5370, 0.5673],\,\tilde{r}_{3} =[0.5711, 0.6232].$$

  • Step 4 Comparing the final decision information using the proposed intuitionistic fuzzy preference relation.

According to Eq. (16), we have

$$\begin{aligned} &\begin{array}{*{20}l} \quad\quad\quad\quad\quad\quad\quad\quad{u_{1} } & \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad{u_{2} } &\quad\quad\quad\quad\quad\quad\quad\quad{u_{3} } \\ \end{array} \hfill \\ P &= \begin{array}{*{20}c} {u_{1} } \\ {u_{2} } \\ {u_{3} } \\ \end{array} \left( {\begin{array}{*{20}c} {\left\langle {0,\;0,\;1} \right\rangle } & {\left\langle {0. 5 3 0 1,\;0. 0 0 6 2,\;0.4 6 3 7} \right\rangle } & {\left\langle {0,\;0.6 5 1 3,\;0.3 4 8 7} \right\rangle } \\ {\left\langle {0. 0 0 6 2,\;0. 5 3 0 1,\;0.4 6 3 7} \right\rangle } & {\left\langle {0,\;0,\;1} \right\rangle } & {\left\langle {0,\; 1,\;0} \right\rangle } \\ {\left\langle {0.6 5 1 3,\;0,\;0.3 4 8 7} \right\rangle } & {\left\langle { 1,\;0,\; 0} \right\rangle } & {\left\langle {0,\;0,\;1} \right\rangle } \\ \end{array} } \right). \hfill \\ \end{aligned}$$

Because there are only three alternatives, we can compare the alternatives directly, i.e.,

$$u_{1} \ge_{0. 5 3 0 1} u_{2},\,u_{3} \ge_{0.651 3} u_{1} \quad {\text{and}} \quad u_{3} \ge_{ 1} u_{2}.$$

Thus, the ranking of alternatives should be \(\lambda_{3} \succ \lambda_{1} \succ \lambda_{2}\).

To illustrate the application of the proposed optimization model, we build the following model according to Eq. (17), i.e.,

$$\begin{aligned} f(\lambda_{1} ,\;\lambda_{2} ,\;\lambda_{3} ) = {\rm min}\sum\limits_{i = 1}^{3} {\sum\limits_{j = i + 1}^{3} {\left( {d_{ij}^{ - } + d_{ij}^{ + } } \right)} } \hfill \\ s.t.\; \left\{ \begin{aligned} 0.5(\lambda_{i} - \lambda_{j} + 1) + d_{ij}^{ - } \ge \mu_{{p(\tilde{r}_{i} \ge \tilde{r}_{j} )}} ; \hfill \\ \mu_{{p(\tilde{r}_{1} \ge \tilde{r}_{2} )}} = 0.4228,\;\mu_{{p(\tilde{r}_{1} \ge \tilde{r}_{3} )}} = 0,\;\mu_{{p(\tilde{r}_{2} \ge \tilde{r}_{3} )}} = 0; \hfill \\ 0.5(\lambda_{i} - \lambda_{j} + 1) - d_{ij}^{ + } \le 1 - v_{{p(\tilde{r}_{i} \ge \tilde{r}_{j} )}} ;\; \hfill \\ v_{{p(\tilde{r}_{1} \ge \tilde{r}_{2} )}} = 0.1039,\;\mu_{{p(\tilde{r}_{1} \ge \tilde{r}_{3} )}} = 0.6851,\;\mu_{{p(\tilde{r}_{2} \ge \tilde{r}_{3} )}} = 0.9941; \hfill \\ \sum\limits_{i = 1}^{3} {\lambda_{i} } = 1,\;\lambda_{i} \ge 0,\;i = 1,\;2,\;3;\; \hfill \\ \;\;\;\;\;\;\;\;d_{ij}^{ - } ,\;d_{ij}^{ + } \ge 0,\;i = 1,\;2,\; \cdots ,\;m,\;j = i + 1,\; \cdots ,\;m. \\ \end{aligned} \right. \hfill \\ \end{aligned}$$

By using Lingo 11.0, the ranking values of all alternatives are listed as follows:

$$\lambda_{1} = 0.0118,\;\lambda_{2} = 0,\;\lambda_{3} = 0.9882.$$

It is worth noting that \(d_{ij}^{ - } = 0,\;d_{ij}^{ + } = 0,\;i = 1,\;2,\;3,\;j = i + 1,\; \cdots ,\;3\).

  • Step 5 Ranking \(\lambda_{i} \left( {i = 1,\;2,\;3} \right)\) in descending order and selecting the best alternative.

According to the results obtained in Step 4, the order of \(\lambda_{i} \left( {i = 1,\;2,\;3} \right)\) is

$$\lambda_{3} \succ \lambda_{1} \succ \lambda_{2}.$$

Thus, the best choice is the third alternative, i.e., \(u_{3}\), which is the same with the direct comparison.

  • Step 6 End

5.3 Comparisons and analysis

For the reason that there exist some other comparing methods, next we make a comparison between our proposed intuitionistic fuzzy possibility degree method and other existed approaches, the results are shown in the following Table 3.

Table 3 Ordering of the strategies
Full size table

Form Table 3, it can be concluded that the result produced by our method is the same as other ranking approaches. The differences among these methodologies can be summarized as follows:

Facchinetti et al. (12) developed a ranking measure for fuzzy triangular numbers, which can be used for ordering interval-valued fuzzy numbers, while Wang et al. (37) finished such extension and gave a preference order for interval-valued fuzzy numbers. But there exists cases that the order relation cannot identify, such as the situation that two intervals are same centered. Jiang, et al. (21) listed all possible locations between two interval-valued fuzzy numbers and proposed corresponding comparison laws, but this method cannot handle the case that one or more intervals degenerate to be a crisp number. In the work of Sun and Yao [35], the possibility degree function proposed by Nakahara et al. [26] is used to give an interval reliability analysis, and the same issue of this method is that it cannot distinguish two intervals with the same centers. Finally, Song et al. [33] tried to get a complete rank of objects and defined the concept of directional distance index (DDI), it’s a pity that the case of equi-centered intervals cannot be completely solved. Different form these methodologies, the developed method in this paper see the equi-centered intervals as equivalent intervals with certain level, and thus one can rank them according to other auxiliary information, for example, the different degrees that two equi-centered intervals are dominated by another interval.

Thus, the proposed ranking method provides a novel perspective to handle the order relation of interval-valued numbers so that a total preference relation among intervals can be obtained.

6 Conclusions

In this study, a direct fuzzy comparing approach for ranking of interval numbers is presented. The main advantage of the intuitionistic fuzzy possibility degree is that it can distinguish the comparable and incomparable parts between any two intervals. It can solve the incomparable cases of nested interval numbers and the intuitive inconsistency starting from the cases of ranking equi-centered interval numbers problems effectively via introducing the hesitant degree in the concept of possibility degree.

The intuitionistic fuzzy possibility degree can be applied in lots of fields such as decision theory, data analysis, artificial intelligence, and socioeconomic systems. In this paper, the application of IFPD in interval number-based multi-attribute decision-making issue regarding the selection of maintainability design is presented. One can see the capability of such approach in handling uncertainties and the simplicity of the proposed IFPD-based methodology in real application.

In future research, there are some interval-valued fuzzy numbers such as the interval-valued intuitionistic fuzzy sets [2, 3, 62], the interval-valued hesitant fuzzy sets [13], interval-valued fuzzy soft sets [45], and some extensions of interval numbers, such as the triangular fuzzy numbers [11] and the trapezoidal fuzzy number [1]. Although there are kinds of approaches for ranking of such fuzzy numbers or fuzzy sets, we also suggest applying the proposed methodology to compare them. Besides, another possible direct fuzzy ways for comparing of fuzzy numbers or fuzzy sets are also suggested.