1 Introduction

Destructive emergency events (EEs) are ubiquitous and often lead to unexpected catastrophic consequences (Wang et al. 2019; Zhang and Li 1967; Shafer 1976) to yield the interval-valued hesitant fuzzy TODIM (IVHF-TODIM) method to handle the EDM problem in which the preference information is given as an IVHFS (Asan et al. 2018; Zeng et al. 2019). Currently, there is no IVHF MADM method that comprehensively considers the DM’s psychological behavior and employs IER to tackle EDM problems. We hope to address this shortcoming using our proposed method.

Specifically, our work contributes to the extant knowledge by:

  1. 1.

    Introducing a function to find the gain and loss degrees based on the geometric area method.

  2. 2.

    Applying the probability density function and geometric area method to specify the gain and loss matrices of the interval belief degrees.

  3. 3.

    Establishing a way to develop the dominance degree matrix to obtain the ranking of the decision alternatives.

The rest of this paper is set as follows: Section 2 introduces the IER and TODIM methods. Section 3 presents an IVHF-TODIM method for solving the EDM problem. A case study of a decision alternative selection issue is used in Sect. 4 to illustrate the novelty and validity of the proposed method. This followed by a comparison with the other extant methods. Section 5 concludes the paper.

2 Preliminaries: concepts and definitions

We introduce the following to this paper as shown below.

Notation

\(\beta_{n,i}^{{}} \left( {a_{l} } \right)\)

Belief degree of alternative \(a_{l}\) evaluated as class \(H_{n}\) under attribute \(e_{i}\)

\(\beta_{H,i}^{{}} \left( {a_{l} } \right)\)

Belief degree assigned to the whole set

\(m_{n,i}\)

Interval basic probability assignment of attribute i

\(\overline{m}_{H,i}\)

Uncertainty caused by the relative importance of attribute i

\(\tilde{m}_{H,i}\)

Uncertainty caused by the incompleteness of belief degree in attribute i

\(D = \left[ {d_{ij} } \right]_{n \times m}\)

Decision-making matrix

\(R = \left[ {r_{ij} } \right]_{n \times m}\)

Normalized decision-making matrix

\(\omega_{jr}\)

Relative weight of attribute \(C_{j}\) to reference attribute \(C_{r}\)

\(\varphi_{iq}^{j}\)

Dominance degree of alternative a \(\tilde{A}_{i}\) over \(\tilde{A}_{q}\) for attribute \(C_{j}\)

\(\phi_{iq}\)

Overall dominance degree of alternative \(\tilde{A}_{i}\) over alternative \(\tilde{A}_{q}\)

\(\xi \left( {\tilde{A}_{i} } \right)\)

Overall dominance value of alternative \(\tilde{A}_{i}\)

\(E = \left\{ {e_{1} , \ldots ,e_{e} } \right\}\)

Set of decision makers, where \(e_{l}\) denotes DM l, l = 1, …,

\(A = \left\{ {A_{1} , \ldots ,A_{n} } \right\}\)

Set of decision alternatives, i = 1,…, n

\(C = \left\{ {C_{1} , \ldots ,C_{m} } \right\}\)

Set of attributes, where \(C_{j}\) is attribute j, \(j = 1, \ldots ,m\)

\(W_{E} = \left\{ {\omega_{e1} , \ldots ,\omega_{ee} } \right\}\)

Weight vector for the experts

\(W_{C} = \left\{ {\omega_{1} ,...,\omega_{m} } \right\}\)

Weight vector for the attributes

\(Z = \left\{ {Z_{1} , \ldots ,Z_{t} } \right\}\)

Set of emergency states

\(\tilde{h}_{A} \left( {x_{i} } \right) = \left\{ {\tilde{\gamma }|\tilde{\gamma } \in \tilde{h}_{A} \left( {x_{i} } \right)} \right\}\)

\(\tilde{h}_{A} \left( {x_{i} } \right)\) (Liu et al. 2019) is an IVHF element, and \(\tilde{\gamma } = \left[ {\tilde{\gamma }^{L} ,\tilde{\gamma }^{U} } \right]\) is an interval number

2.1 Interval evidential reasoning (IER)

In this paper, the ER approach using interval belief degrees (Wang et al. 2006) is labelled as the IER approach, which will allow both quantitative and qualitative attributes to be modeled using interval data. This is a generalization of the Dempster–Shafer theory (Dempster 1967; Shafer 1976).

Definition 1

(Wang et al. 2006). Suppose \(S\left( {e_{i} \left( {a_{l} } \right)} \right) = \left\{ {\left( {H_{n} ,\left[ {\beta_{n,i}^{ - } \left( {a_{l} } \right),\beta_{n,i}^{ + } \left( {a_{l} } \right)} \right]} \right),\quad n = 1, \ldots ,N} \right\}\) is an incomplete interval-valued distribution evaluation vector. Then, the interval belief degree, \(\beta_{H,i} \left( {a_{l} } \right)\), assigned to the set \(H\) can be expressed as

$$ \left\{ \begin{array}{*{20}l} \beta_{H,i}^{ - } \left( {a_{l} } \right) = \max \left( {0,1 - \sum\limits_{n = 1}^{N} {\beta_{n,i}^{ + } \left( {a_{l} } \right)} } \right) \hfill \\ \beta_{H,i}^{ + } \left( {a_{l} } \right) = 1 - \sum\limits_{n = 1}^{N} {\beta_{n,i}^{ - } \left( {a_{l} } \right)} \hfill \\ \end{array} \right. $$
(1)

Definition 2

(Wang et al. 2006). Let \(y_{i} \in \left[ {y_{i}^{ - } ,y_{i}^{ + } } \right]\) be an interval number, which contains one or more assessment levels. Without any loss of generality, consider a situation with two evaluation levels \(Y_{n,i}\) and \(Y_{n + 1,i}\) in the interval \(\left[ {y_{i}^{ - } ,y_{i}^{ + } } \right]\) (see Fig. 1).

Fig. 1
figure 1

Relationship among interval values and evaluation grades (\(y_{i}\) includes \(Y_{n,i}\) and \(Y_{n + 1,i}\))

Full size image

Let \(\beta_{n - 1,i} \in \left[ {\beta_{n - 1,i}^{ - } ,\beta_{n - 1,i}^{ + } } \right]\), \(\beta_{n,i} \in \left[ {\beta_{n,i}^{ - } ,\beta_{n,i}^{ + } } \right]\), \(\beta_{n + 1,i} \in \left[ {\beta_{n + 1,i}^{ - } ,\beta_{n + 1,i}^{ + } } \right]\), and \(\beta_{n + 2,i} \in \left[ {\beta_{n + 2,i}^{ - } ,\beta_{n + 2,i}^{ + } } \right]\) be the interval belief degrees in which \(y_{i} \in \left[ {y_{i}^{ - } ,y_{i}^{ + } } \right]\) may be assessed. \(Y_{n - 1,i}\), \(Y_{n,i}\), \(Y_{n + 1,i}\) and \(Y_{n + 2,i}\) are interval numbers. As \(y_{i}\) can only lie in one of the three intervals, namely \(\left[ {y_{i}^{ - } ,Y_{n,i} } \right)\), \(\left[ {Y_{n,i} ,Y_{n + 1,i} } \right]\) and \(\left[ {Y_{n,i} ,Y_{n + 1,i} } \right]\), we need to introduce the following 0–1 binary variables:

$$ I_{k - 1.k} = \left\{ {\begin{array}{ll} 1 & {y_{i} \in \left[ {Y_{k - 1,i} ,Y_{k,i} } \right]} \\ 0 & {{\text{otherwise}}} \\ \end{array} } \right.\quad k = n,n + 1,n + 2 $$
(2)

The interval belief degrees are presented as follows:

$$ \beta_{n - 1,i}^{ - } = 0\;{\text{and}}\;\beta_{n - 1,i}^{ + } = \frac{{y_{i}^{ - } - Y_{n - 1,i} }}{{Y_{n,i} - Y_{n - 1,i} }} \cdot I_{n - 1,n} $$
(3)
$$ \beta_{n,i}^{ - } = 0\;{\text{and}}\;\beta_{n,i}^{ + } = I_{n - 1,n} + I_{n,n + 1} $$
(4)
$$ \beta_{n + 1,i}^{ - } = 0\;{\text{and}}\;\beta_{n + 1,i}^{ + } = I_{n,n + 1} + I_{n + 1,n + 2} $$
(5)
$$ \beta_{n + 2,i}^{ - } = 0\;{\text{and}}\;\beta_{n + 2,i}^{ + } = \frac{{y_{i}^{ + } - Y_{n + 1,i} }}{{Y_{n + 2,i} - Y_{n + 1,i} }} \cdot I_{n + 1,n + 2} $$
(6)

We have \(I_{n - 1,n} + I_{n,n + 1} + I_{n + 1,n + 2} = 1\), and only one of 0–1 binary variables are nonzero.

For the sake of aggregating the multiple interval belief structures correctly, the ER nonlinear optimization models are formed.

First, through the weights \(\omega_{i}\) and interval belief degrees, the interval belief degrees are transformed into interval probability mass functions (m) using the following relations:

$$ m_{n,i} = m_{i} \left( {H_{n} } \right) \in \left[ {m_{n,i}^{ - } ,m_{n,i}^{ + } } \right] = \left[ {\omega_{i} \beta_{n,i}^{ - } \left( {a_{l} } \right),\omega_{i} \beta_{n,i}^{ + } \left( {a_{l} } \right)} \right],\quad n = 1, \ldots ,N;\quad i = 1, \ldots ,L, $$
(7)
$$ \overline{m}_{H,i} = \overline{m}_{i} (H) = 1 - \omega_{i} ,i = 1, \ldots ,L, $$
(8)
$$ \tilde{m}_{H,i} = \tilde{m}_{i} (H) \in \left[ {\tilde{m}_{H,i}^{ - } ,\tilde{m}_{H,i}^{ + } } \right] = \left[ {\omega_{i} \beta_{H,i}^{ - } \left( {a_{l} } \right),\omega_{i} \beta_{H,i}^{ + } \left( {a_{l} } \right)} \right],\quad i = 1, \ldots ,L, $$
(9)

where \(\sum\nolimits_{n = 1}^{N} {m_{n,i} } + \overline{m}_{H,i} + \tilde{m}_{H,i} = 1\) for \(i = 1\) to \(L\) and \(\sum\nolimits_{i = 1}^{L} {\omega_{i} } = 1\).

Next, from the nonlinear optimization models, the interval probability masses on \(L\) basic attributes are aggregated and transformed into overall interval belief degrees for each n, \(n = 1, \ldots ,N\):

$$ {\text{max/min}}\quad \beta_{n} \left( {a_{l} } \right) = \frac{{m_{n} }}{{1 - \overline{m}_{H} }} $$
(10)
$$ {\rm{s}}.{\rm{t}}{.}\quad m_{n} = k\left[ {\prod\limits_{i = 1}^{L} {\left( {m_{n,i} + \overline{m}_{H,i} + \tilde{m}_{H,i} } \right) - \prod\limits_{i = 1}^{L} {\left( {\overline{m}_{H,i} + \tilde{m}_{H,i} } \right)} } } \right],\quad n = 1, \ldots ,N $$
(11)
$$ \tilde{m}_{H} = k\left[ {\prod\limits_{i = 1}^{L} {\left( {\overline{m}_{H,i} + \tilde{m}_{H,i} } \right) - \prod\limits_{i = 1}^{L} {\overline{m}_{H,i} } } } \right] $$
(12)
$$ \overline{m}_{H} = k\left[ {\prod\limits_{i = 1}^{L} {\overline{m}_{H,i} } } \right] $$
(13)
$$ k = \left[ {\sum\limits_{n = 1}^{N} {\prod\limits_{i = 1}^{L} {\left( {m_{n,i} + \overline{m}_{H,i} + \tilde{m}_{H,i} } \right) - \left( {N - 1} \right)\prod\limits_{i = 1}^{L} {\left( {\overline{m}_{H,i} + \tilde{m}_{H,i} } \right)} } } } \right]^{ - 1} $$
(14)
$$ m_{n,i}^{ - } \le m_{n,i} \le m_{n,i}^{ + } ,\quad n = 1, \ldots ,N;\quad i = 1, \ldots ,L $$
(15)
$$ \overline{m}_{H,i} = 1 - \omega_{i} \;{\text{and}}\;\tilde{m}_{H,i}^{ - } \le \tilde{m}_{H,i} \le \tilde{m}_{H,i}^{ + } ,\quad i = 1, \ldots ,L $$
(16)
$$ \sum\limits_{n = 1}^{N} {m_{n,i} + \overline{m}_{H,i} + } \tilde{m}_{H,i} = 1,\quad i = 1, \ldots ,L $$
(17)

where \(\beta_{n}^{ - } \left( {a_{l} } \right)\) and \(\beta_{n}^{ + } \left( {a_{l} } \right)\) are the optimal objective function values of the above nonlinear optimization model, which constitutes an overall interval belief degree \(\left[ {\beta_{n}^{ - } \left( {a_{l} } \right),\beta_{n}^{ + } \left( {a_{l} } \right)} \right]\). The aggregated assessment is denoted by \(S\left( {y\left( {a_{l} } \right)} \right) =\) \(\left\{ {\left( {H_{n} ,\left[ {\beta_{n}^{ - } \left( {a_{l} } \right),\beta_{n}^{ + } \left( {a_{l} } \right)} \right]} \right)} \right\}\), \(n = 1, \ldots ,N\). When \(\beta_{H} \left( {a_{l} } \right) = \frac{{\tilde{m}_{H} }}{{1 - \overline{m}_{H} }}\) is the objective function instead of \(\beta_{n} \left( {a_{l} } \right) = \frac{{m_{n} }}{{1 - \overline{m}_{H} }}\), we obtain the interval belief degree for \(\beta_{H} \left( {a_{l} } \right)\), which can be written as \(\left[ {\beta_{H}^{ - } \left( {a_{l} } \right),\beta_{H}^{ + } \left( {a_{l} } \right)} \right]\).

2.2 Classical TODIM method

The classical TODIM is an MADM method proposed by Gomes and Lima (1992), based on prospect theory. The gist of the classical TODIM is to establish a dominance degree of each alternative over the other alternatives based on the value function of prospect theory.

The steps of the TODIM method are as follows.

Step 2.2.1 Normalize the decision matrix \(D = \left[ {d_{ij} } \right]_{n \times m}\) into matrix \(R = \left[ {r_{ij} } \right]_{n \times m}\).

Step 2.2.2 Compute the relative weight \(\omega_{jr}\) of attribute \(C_{j}\) to the reference attribute \(C_{r}\), \(\omega_{jr} = \omega_{j} /\omega_{r}\), where \(\omega_{r} = \max \left\{ {\omega_{j} |j,r = 1, \ldots ,m} \right\}\).

Step 2.2.3 The dominance degree \(\varphi_{iq}^{j}\) of alternative \(\tilde{A}_{i}\) over alternative \(\tilde{A}_{q}\) for attribute \(C_{j}\) is given by

$$ \varphi_{iq}^{j} = \left\{ {\begin{array}{ll} {\sqrt {\frac{{\left( {r_{ij} - r_{qj} } \right)\omega_{jr} }}{{\sum\nolimits_{j = 1}^{m} {\omega_{jr} } }},} } & {r_{ij} - r_{qj} > 0} \\ {0,} & {r_{ij} - r_{qj} = 0} \\ { - \frac{1}{\theta }\sqrt {\frac{{\left( {r_{qj} - r_{ij} } \right)\sum\nolimits_{j = 1}^{m} {\omega_{jr} } }}{{\omega_{jr} }}} ,} & {r_{ij} - r_{qj} < 0} \\ \end{array} } \right. $$
(18)

where \(\theta\) is the attenuation coefficient of the losses.

If \(r_{ij} - r_{qj} > \left( < \right)0\), then \(r_{ij} - r_{qj}\) represents the gain (loss) of alternative \(\tilde{A}_{i}\) over alternative \(\tilde{A}_{q}\) for attribute \(C_{j}\), respectively.

Step 2.2.4 The comprehensive dominance degree \(\phi_{iq}\) of alternative \(\tilde{A}_{i}\) over alternative \(\tilde{A}_{q}\) is found from \(\phi_{iq} = \sum\nolimits_{j = 1}^{m} {\phi_{iq}^{j} } (i,q = 1, \ldots ,n)\).

Step 2.2.5 The overall dominance value \(\xi \left( {\tilde{A}_{i} } \right)\) of alternative \(\tilde{A}_{i}\) is obtained using

$$ \xi \left( {\tilde{A}_{i} } \right) = \frac{{\sum\nolimits_{q = 1}^{m} {\phi_{iq} } - \min_{i} \left\{ {\sum\nolimits_{q = 1}^{m} {\phi_{iq} } } \right\}}}{{\max_{i} \left\{ {\sum\nolimits_{q = 1}^{m} {\phi_{iq} } } \right\} - \min_{i} \left\{ {\sum\nolimits_{q = 1}^{m} {\phi_{iq} } } \right\}}} $$
(19)

Step 2.2.6 The rank order of the alternatives is formed using Eq. (19). The larger the \(\xi \left( {\tilde{A}_{i} } \right)\)is, the better alternative \(\tilde{A}_{i}\) will be.

3 IVHF-TODIM method based on IER

We now present an IVHF-TODIM method, which is different from the existing methods (Asan et al. 2018; Xue et al. 2019; Zhou et al. 2018; Zhang 2000), to address the EDM problem by including a DM’s psychological behavior.

3.1 Gain and loss functions based on geometric area method

Using the geometric area method, the gain matrix and loss matrix of the interval belief degrees are set. Let \(\beta_{rk}^{{}} \left( {A_{i} } \right) = \left[ {\beta_{rk}^{ - } \left( {A_{i} } \right),\beta_{rk}^{ + } \left( {A_{i} } \right)} \right]\) be the aggregated result of decision alternative \(A_{i}\), given that emergency state \(Z_{k}\) has occurred concerning the evaluation level \(m_{s} (s = 1, \ldots ,r)\) as obtained from Eqs. (10)–(17).

By aggregating the results of the alternatives, the gain (loss) matrix for each interval belief degree can be constructed by measuring the gain (loss) degree of one interval belief degree relative to another, respectively. To solve for the interval belief degrees, we need to clarify the positional relationship of these interval belief degrees (see Table 1).

Table 1 Positional relationship between \(\beta_{{r_{1} k}} \left( {A_{i} } \right)\) and \(\beta_{{r_{2} k}} \left( {A_{j} } \right)\)
Full size table

Definition 3.

Suppose \(\beta_{{r_{1} k}} \left( {A_{i} } \right){ = }\left[ {\beta_{{r_{1} k}}^{ - } \left( {A_{i} } \right),\quad \beta_{{r_{1} k}}^{ + } \left( {A_{i} } \right)} \right]\) and \(\beta_{{r_{2} k}} (A_{q} ){ = }\left[ {\beta_{{r_{2} k}}^{ - } (A_{q} ),\quad \beta_{{r_{2} k}}^{ + } (A_{q} )} \right]\) are arbitrary interval belief degrees, \(S_{{\beta_{{r_{1} k}} \left( {A_{i} } \right)}}\) and \(S_{{\beta_{{r_{2} k}} \left( {A_{q} } \right)}}\) denote the area of \(\beta_{{r_{1} k}} \left( {A_{i} } \right)\) and \(\beta_{{r_{2} k}} (A_{q} )\), respectively, and \(S_{{\beta_{{r_{1} k}} \left( {A_{i} } \right)}}^{1}\) and \(S_{{\beta_{{r_{2} k}} (A_{q} )}}^{2}\) denote the non-overlap** part that belongs to \(\beta_{{r_{1} k}} \left( {A_{i} } \right)\) and \(\beta_{{r_{2} k}} (A_{q} )\), respectively. Then, \(S\) is the area of the intersection of two interval belief degrees. Let \(g_{iqs}^{k}\) (\(l_{iqs}^{k}\)) and be the gain (loss) degree of decision alternative \(A_{i}\) relative to \(A_{q}\), given that emergency state \(Z_{k}\) has occurred under evaluation level \(m_{s}\), respectively. The gain (and loss) degree \(g_{iqs}^{k}\) (and \(l_{iqs}^{k}\)) of \(A_{i}\) relative to \(A_{q}\), respectively, can be expressed as

$$ g_{iqs}^{k} \left( {\beta_{{r_{1} k}} \left( {A_{i} } \right) \ge \beta_{{r_{2} k}} \left( {A_{q} } \right)} \right) = \frac{1}{2}\left( {\frac{{S_{{\beta_{{r_{1} k}} \left( {A_{i} } \right)}}^{1} + S/2}}{{S_{{\beta_{{r_{1} k}} \left( {A_{i} } \right)}} }} + \frac{{S_{{\beta_{{r_{2} k}} (A_{q} )}}^{2} + S/2}}{{S_{{\beta_{{r_{2} k}} \left( {A_{q} } \right)}} }}} \right) $$
(20)
$$ l_{iqs}^{k} \left( {\beta_{{r_{1} k}} \left( {A_{i} } \right) < \beta_{{r_{2} k}} (A_{q} )} \right) = 1 - g_{iqs}^{k} \left( {\beta_{{r_{1} k}} \left( {A_{i} } \right) \ge \beta_{{r_{2} k}} \left( {A_{q} } \right)} \right) $$
(21)

The areas of \(S_{{\beta_{{r_{1} k}} \left( {A_{i} } \right)}}\), \(S_{{\beta_{{r_{2} k}} \left( {A_{q} } \right)}}\), \(S_{{\beta_{{r_{1} k}} \left( {A_{i} } \right)}}^{1}\), \(S_{{\beta_{{r_{2} k}} (A_{q} )}}^{2}\), and \(S\) are denoted by the probability density function \(f\left( x \right)\). Next, we use the following definition.

Definition 3

Fan and Liu 2010). Let \(x\) be a variable in the interval belief degree \(\beta_{rk}^{{}} \left( {A_{i} } \right) =\) \(\left[ {\beta_{rk}^{ - } \left( {A_{i} } \right),\beta_{rk}^{ + } \left( {A_{i} } \right)} \right]\), \(\beta_{rk}^{ - } \left( {A_{i} } \right) \le x \le \beta_{rk}^{ + } \left( {A_{i} } \right)\). The probability density function of \(x\) is stated as:

$$ f\left( x \right) = \left\{ {\begin{array}{*{20}l} {\frac{1}{b - a},} & {a \le x \le b,} \\ {0,} & {{\text{otherwise}}} \\ \end{array} } \right. $$
(22)

where \(\int_{a}^{b} {f\left( x \right)} = 1\) and \(f\left( x \right) \ge 0\) for all \(x \in \left[ {a,b} \right]\).

Table 2 contains the expressions used to obtain the gains and losses for the interval belief degrees for the cases stated in Table 1. The steps are detailed in “Appendix”.

Table 2 Gains and losses for all possible cases
Full size table

Based on the gain (and loss) degree of alternative \(A_{i}\) to \(A_{q}\), the gain matrix \(g_{s}^{k} = \left[ {g_{iqs}^{k} } \right]_{n \times n}\) and loss matrix \(l_{s}^{k} = \left[ {l_{iqs}^{k} } \right]_{n \times n}\), given that emergency state \(Z_{k}\) has occurred under evaluation level \(m_{s}\) is thus

$$ g_{s}^{k} = \begin{array}{*{20}c} {\begin{array}{*{20}c} {A_{1} } & { \, A_{2} } & \ldots & {A_{n} } \\ \end{array} } \\ {\begin{array}{*{20}c} {A_{1} } \\ {A_{2} } \\ \vdots \\ {A_{n} } \\ \end{array} \, \left[ {\begin{array}{*{20}c} {g_{11s}^{k} } & {g_{12s}^{k} } & \cdots & {g_{1ns}^{k} } \\ {g_{21s}^{k} } & {g_{22s}^{k} } & \cdots & {g_{2ns}^{k} } \\ \vdots & \vdots & \vdots & \vdots \\ {g_{n1s}^{k} } & {g_{n2s}^{k} } & \cdots & {g_{nns}^{k} } \\ \end{array} } \right],} \\ \end{array} \quad k = 1, \ldots t;\;s = 1, \ldots ,r $$
(23)
$$ l_{s}^{k} = \begin{array}{*{20}c} {\begin{array}{*{20}c} {A_{1} } & { \, A_{2} } & { \, ...} & {A_{n} } \\ \end{array} } \\ {\begin{array}{*{20}c} {A_{1} } \\ {A_{2} } \\ \vdots \\ {A_{n} } \\ \end{array} \, \left[ {\begin{array}{*{20}c} {l_{11s}^{k} } & {l_{12s}^{k} } & \cdots & {l_{1ns}^{k} } \\ {l_{21s}^{k} } & {l_{22s}^{k} } & \cdots & {l_{2ns}^{k} } \\ \vdots & \vdots & \vdots & \vdots \\ {l_{n1s}^{k} } & {l_{n2s}^{k} } & \cdots & {l_{nns}^{k} } \\ \end{array} } \right]} \\ \end{array} ,\quad k = 1, \ldots ,t;\;s = 1, \ldots ,r $$
(24)

where \(g_{iis}^{k} = l_{iis}^{k} = 0\) for all \(i \in N\).

To facilitate analysis and computation, it is necessary to normalize the gain and loss matrices. Therefore, the terms \(\overline{g}_{iqs}^{k}\) and \(\overline{l}_{iqs}^{k}\) can be obtained from

$$ \overline{g}_{iqs}^{k} = \frac{{g_{iqs}^{k} }}{{g_{ks}^{\max } }},\quad i,q = 1, \ldots ,n;\quad k = 1, \ldots ,t;\quad s = 1, \ldots r $$
(25)
$$ \overline{l}_{iqs}^{k} = \frac{{l_{iqs}^{k} }}{{l_{ks}^{\max } }},\quad i,q = 1, \ldots ,n;\quad k = 1, \ldots t;\quad s = 1, \ldots r $$
(26)

3.2 Dominance degree matrix

Prior studies assume that the DMs are often bounded rational under risk and uncertainty, and their behavior influences the decision-making process. We extend prospect theory to find the relative gains and losses of two alternatives under the evaluation level, to better reflect reality. The dominance degree is obtained as

$$ \varphi_{iqs}^{k} = \left\{ {\begin{array}{*{20}l} {\varphi_{iqs}^{k( + )} = \sqrt {\left( {\overline{g}_{iqs}^{k} } \right)^{\alpha } } ,} & {\beta_{{r_{1} k}} \left( {A_{i} } \right) \ge \beta_{{r_{2} k}} \left( {A_{q} } \right)} \\ {\varphi_{iqs}^{k( - )} = - \frac{1}{\theta }\sqrt { - \lambda \left( { - \overline{l}_{iqs}^{k} } \right)^{\beta } } ,} & {\beta_{{r_{1} k}} \left( {A_{i} } \right) < \beta_{{r_{2} k}} \left( {A_{q} } \right)} \\ \end{array} } \right. $$
(27)

where \(\alpha\) and \(\beta\) are the power parameters related to the gains and losses, and \(\alpha\) is a concave degree parameter related to gains, and \(\beta\) is convex degree parameter related to losses, respectively; \(\lambda\) denotes the risk-aversion parameter, \(0 \le \alpha ,\;\beta \le 1,\;\lambda > 1\), and \(\theta\) is an attenuation factor of the loss. If θ > (<) 1, then the influence of loss will decrease (increase). In this paper, \(\theta = 1\). Following from previous studies such as Tversky and Kahneman (1992) who used \( \, \alpha = 0.89,\) \( \beta = 0.92\) \( \, \lambda = 2.25\); Abdellaoui et al. (2007) applied \(\alpha = 0.725\), \(\beta = 0.717\), \(\lambda = 2.04\); Liu et al. (2014) used \( \, \alpha = 0.85\), \(\beta = 0.85\), \(\lambda = 4.1\). In this paper, we choose the parameter values of Simon (1955).

3.3 Comprehensive dominance value

Furthermore, the overall dominance degree of alternative \(A_{i}\) over alternative \(A_{q}\), given that emergency state \(Z_{k}\) has occurred, is expressed by

$$\Phi _{iq}^{k} = \sum\limits_{s = 1}^{r} {\varphi_{iqs}^{k( + )} } + \sum\limits_{s = 1}^{r} {\varphi_{iqs}^{k( - )} } $$
(28)

Thus, the comprehensive dominance degree matrix is as follows

$$\Phi _{k} { = }\left[ {\Phi _{iq}^{k} } \right]_{n \times n} { = }\begin{array}{*{20}c} {\begin{array}{*{20}c} {A_{1} } & {A_{2} } & \ldots & {A_{n} } \\ \end{array} } \\ {\begin{array}{*{20}c} {A_{1} } \\ {A_{2} } \\ \vdots \\ {A_{n} } \\ \end{array} \, \left[ {\begin{array}{*{20}c} {\Phi _{11}^{k} } & {\Phi _{12}^{k} } & \cdots & {\Phi _{1n}^{k} } \\ {\Phi _{21}^{k} } & {\Phi _{22}^{k} } & \cdots & {\Phi _{2n}^{k} } \\ \vdots & \vdots & \vdots & \vdots \\ {\Phi _{n1}^{k} } & {\Phi _{n2}^{k} } & \cdots & {\Phi _{nn}^{k} } \\ \end{array} } \right]} \\ \end{array} ,\quad k = 1, \ldots ,t. $$
(29)

According to \(\Phi _{k}\), the comprehensive dominance value of alternative \(A_{i}\) over alternative \(A_{q}\), given that emergency state \(Z_{k}\) has occurred, is expressed by

$$ \xi^{k} \left( {A_{i} } \right) = \frac{{\psi^{k} \left( {A_{i} } \right) - \mathop {\min }\limits_{i \in N} \left\{ {\psi^{k} \left( {A_{i} } \right)} \right\}}}{{\mathop {\max }\limits_{i \in N} \left\{ {\psi^{k} \left( {A_{i} } \right)} \right\} - \mathop {\min }\limits_{i \in N} \left\{ {\psi^{k} \left( {A_{i} } \right)} \right\}}},\quad i \in N. $$
(30)

where \(\psi^{k} \left( {A_{i} } \right) = \sum\nolimits_{q = 1}^{n} {\Phi _{iq}^{k} }\).

To facilitate the solution, the following algorithm is proposed.

  • Step 1 Establish the framework for assessing the EDM.

  • Step 2 The DMs provide individual preference information on the decision alternatives.

  • Step 3 Aggregate the various evaluation information from the DMs using the IVHF-weighted geometric (IVHFWG) operator in Chen et al. 2013.

  • Step 4 Aggregate the evaluation level information from the decision alternatives using Eqs. (2)–(17).

    • Step 4.1 Obtain the interval belief degrees using Eqs. (2)–(6).

    • Step 4.2 Convert the interval belief degrees into interval probability masses using Eqs. (7)–(9).

    • Step 4.3 Aggregate and transform the interval probability masses into the overall interval belief degrees using Eqs. (10)–(17).

  • Step 5 Obtain the gains and losses using Tables 1 and 2.

  • Step 6 Obtain the dominance degree matrix using Eq. (27).

  • Step 7 Find the overall dominance degree for ranking the decision alternatives using Eqs. (28)–(30).

4 Numerical example and comparison

To validate the proposed method, we provide an example of selecting an emergency decision alternative for a massive bushfire. Then, we perform a comparative analysis to illustrate the novelty and superiority of our method compared to some existing methods.

4.1 Emergency decision making for large bushfire

A massive regional bushfire has occurred in summer, threatening the lives and properties of many people and native animals in the vicinity. A certain highway is labelled as the line of containment to control the spread of the bushfire. Because of the dry and unpredictable wind conditions, the fire can transition to three states: (1) the fire spreads to the highway and onto the city; (2) the fire engulfs the farms and homes in the vicinity; (3) heavy rain occurs in the vicinity, which can douse the fire. To obtain the evaluation information of this large bushfire, this paper adopts the methods of expert surveys and questionnaires to obtain the decision alternatives and attribute values. The expert panel consists of three stakeholder groups. Stakeholder group \(D_{1}\) comprises experts from regional government, stakeholder group \(D_{2}\) is composed of the experts from resident communities, and stakeholder group \(D_{3}\) comprises the experts from the regional firefighting services. In this paper, the role of the expert panel is to evaluate the qualitative evaluation attributes of each crisis solution alternative and provide the evaluation values in the form of an IVHFS for a better selection of the optimal choices to solve the bushfire crisis. Three emergency decision alternatives \(A{ = }\left\{ {A_{1} ,A_{2} ,A_{3} } \right\}\) are given by three stakeholder groups for each state, namely

  • \(A_{1} :\) Close the highway completely, launch large-scale firefighting to contain the fire;

  • \(A_{2} :\) Close the highway partially, introduce controlled firefighting to contain the fire, and evacuate people and animals using what is left of the highway;

  • \(A_{3} :\) Leave the highway open, continue to fight the fire using the region’s emergency fire services, and send out fire advisories to the residents in the vicinity to be on the alert.

Each decision alternative has five attributes \(C = \left\{ {C_{1} ,C_{2} ,C_{3} ,C_{4} ,C_{5} } \right\}\), with \(C_{1}\): spread of the fire, \(C_{2}\): weather, \(C_{3}\): size of the bushland, \(C_{4}\): topography, and \(C_{5}\): social condition. The weights of the DMs are assigned a priori as \(W_{E} = \left\{ {0.2722,0.2561,0.4717} \right\}\). The DMs express their evaluations using the IVHFS found in Tables 3, 4, and 5. Table 6 shows the assessment level for each attribute, as revised by the DMs.

Table 3 Decision matrix from DMs under state \(Z_{1}\)
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Table 4 Decision matrix for DMs under state \(Z_{2}\)
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Table 5 Decision matrix for DMs under state \(Z_{3}\)
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Table 6 Assessment level for attribute Cj
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Using the weight vector \(W_{E}\), the overall information is obtained by aggregating the evaluation information from the experts using the IVHFWG operator in Chen et al. (2013) (see Table 7). For example, the value of alternative \(A_{1}\) in state \(Z_{1}\) with respect to attribute \(C_{1}\) is obtained as: \(0.8\wedge 0.2722 \times 0.7\wedge 0.2561 \times 0.6\wedge 0.4717 \approx 0.68\), \(0.9\wedge 0.2722 \times 0.8\wedge 0.2561 \times 0.7\wedge 0.4717 \approx 0.78\).

Table 7 Aggregated values from DMs for state Zi
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Suppose the attribute weights are \(\left( {\omega_{1} ,\omega_{2} ,\omega_{3} ,\omega_{4} ,\omega_{5} } \right)^{\rm T} = \left( {0.274,0.077,0.075,0.281,0.293} \right)^{{\text{T}}}\). Aggregate the evaluation level information from the decision alternatives using Eqs. (5)–(12) as follows:

First, aggregate IVHF numbers as interval-valued numbers (IVNs) for the three states, as shown in Tables 8, 9, and 10 (e.g., the value of alternative \(A_{1}\) in state \(Z_{1}\) with respect to attribute \(C_{1}\) is obtained as \(0.68\wedge 0.27 \times 0.28\wedge 0.27 = 0.63\)):

Table 8 Aggregated IVHFNs into IVNs under state \(Z_{1}\)
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Table 9 Aggregated IVHFNs into IVNs under state \(Z_{2}\)
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Table 10 Aggregated IVHFNs into IVNs under state \(Z_{3}\)
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Using Step 4.1, the interval belief degrees can be obtained as Tables 11, 12, and 13, with the help of Table 6. For instance, take the interval belief degree of \(A_{1}\) under state \(Z_{1}\) with respect to \(C_{1}\). The evaluation value of \(C_{1} A_{1}\) after aggregating is an interval value [0.63, 0.72], which lies between assessment levels G and V. The interval belief degrees, on which \(C_{1} A_{1}\) is assessed to lie between G and V, can be found as follows:

$$ \beta_{3,1}^{ - } \left( {A_{1} } \right) = \frac{0.8 - 0.72}{{0.8 - 0.6}} = 0.38\;\text{and}\;\beta_{3,1}^{ + } \left( {A_{1} } \right) = \frac{0.8 - 0.63}{{0.8 - 0.6}} = 0.84 $$
$$ \beta_{4,1}^{ - } \left( {A_{1} } \right){ = }\frac{0.63 - 0.6}{{0.8 - 0.6}}{ = }0.16\;\text{and}\;\beta_{4,1}^{ + } \left( {A_{1} } \right){ = }\frac{0.72 - 0.6}{{0.8 - 0.6}}{ = }0.62 $$
Table 11 Interval belief degrees under state \(Z_{1}\)
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Table 12 Interval belief degrees under state \(Z_{2}\)
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Table 13 Interval belief degrees under state \(Z_{3}\)
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Therefore, \(C_{1} A_{1}\) under state \(Z_{1}\) can be modeled by using an interval belief structure as {(G, [0.38, 0.84]), (V, [0.16, 0.62])}.

Next, using Step 4.2, convert the interval belief degrees to interval probability masses using Eqs. (7)–(9). For instance, for \(C_{1} A_{1}\) under state \(Z_{1}\), we have

$$ m_{1,1} = m_{2,1} = m_{5,1} = 0 $$
$$ m_{3,1} = \left[ {m_{3,1}^{ - } ,m_{3,1}^{ + } } \right] = \omega_{1} \times \left[ {\beta_{3,1}^{ - } ,\beta_{3,1}^{ + } } \right] = 0.274 \times \left[ {0.38,0.84} \right] = \left[ {0.105,0.231} \right] $$
$$ m_{4,1} = \left[ {m_{4,1}^{ - } ,m_{4,1}^{ + } } \right] = \omega_{1} \times \left[ {\beta_{4,1}^{ - } ,\beta_{4,1}^{ + } } \right] = 0.274 \times \left[ {0.16,0.62} \right] = \left[ {0.043,0.169} \right] $$
$$ \overline{m}_{H,1} = \overline{m}_{1} \left( H \right) = 1 - \omega_{1} = 1 - 0.27 = 0.73 $$
$$ \tilde{m}_{H,1} = \tilde{m}_{1} (H) \in \left[ {\tilde{m}_{H,1}^{ - } ,\tilde{m}_{H,1}^{ + } } \right] = \left[ {\omega_{1} \beta_{H,1}^{ - } \left( {a_{l} } \right),\omega_{1} \beta_{H,1}^{ + } \left( {a_{l} } \right)} \right] = [0.27 \times 0,0.27 \times 0.46] = [0,0.13] $$
$$ \left\{ \begin{array}{*{20}l} \beta_{H,1}^{ - } \left( {a_{l} } \right) = \max \left( {0,1 - \sum\limits_{n = 1}^{N} {\beta_{n,1}^{ + } \left( {a_{l} } \right)} } \right) = \max \left( {0,1 - 0.84 - 0.62} \right) = 0 \hfill \\ \beta_{H,i}^{ + } \left( {a_{l} } \right) = 1 - \sum\limits_{n = 1}^{N} {\beta_{n,i}^{ - } \left( {a_{l} } \right) = 1 - 0.38 - 0.16 = 0.46} \hfill \\ \end{array} \right. $$

Using Step 4.3, aggregate and transform the interval probability masses to the overall interval belief degrees using Eqs. (10)–(17). Table 14 shows the aggregated interval-valued distribution assessment for each alternative.

Table 14 Aggregated interval-valued belief degrees for each decision alternative
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Based on the positional relationship between each of the interval belief degrees shown in Table 1, the gains and losses can be computed according to Tables 1 and 2. In the interest of space, we will list only the results of \(Z_{1}\) as shown in Table 15.

Table 15 Gain and loss values for each decision alternative under state \(Z_{1}\)
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The dominance value of alternative \(A_{i}\) over alternative \(A_{q}\), given that state \(Z_{k}\) occurs for attribute \(C_{j}\) can be obtained from Eq. (27) (see Table 16).

Table 16 Dominance value for each decision alternative under state \(Z_{1}\)
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The overall dominance degree is found from Eqs. (28)–(30), and Table 17 shows the ranking.

Table 17 Overall dominance degree values and corresponding ranking
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From Table 17, it is apparent that the decision alternatives are state dependent. The optimal decision alternatives for states \(Z_{1}\), \(Z_{2}\), and \(Z_{3}\) are \(A_{1}\), \(A_{3}\), and \(A_{2}\), respectively. Clearly, as the bushfire is highly dependent on the realization of the states which can change rapidly over time, it is preferable to offer a suite of “good” solutions for the DMs to consider rather than to point to a single “optimal” solution as suggested by the extant literature. Flexibility during an emergency response to mega disasters is needed especially when dealing with the capriciousness of such states as the recent incident of emergency events in many actual life-threatening bushfires, e.g., China and Australia have testified. The outcome of this numerical example validates the proposed method.

4.2 Comparison analysis

To highlight the effectiveness of the proposed method, this section compares the proposed method with two existing methods, namely

  1. (1)

    The proposed method is compared with a group-based method based on prospect theory (Wang et al. 2017), which excludes the dynamic factors.

  2. (2)

    The proposed approach is compared with a distance-based group decision-making method (Yu and Lai 2011), which ignores the psychological behavior of the DMs.

Table 18 provides the rankings of the three emergency alternatives based on the three states.

Table 18 Comparison results between proposed method and other methods
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From Table 18, when the proposed method is compared against the group method based on prospect theory (Wang et al. 2017) and with a distance-based group decision-making method (Yu and Lai 2011), there is a difference in the outcomes. There are some reasons for this difference:

  1. (1)

    The group-based method, based on prospect theory (Wang et al. 2017), does not take into account the nature of emergency events. In reality, emergency decisions should be dynamically adjusted to improve the emergency response levels as bushfire emergencies can rapidly transition between states in a short time window of hours, as more information from the field is collected in that time. So, decision making must reflect that nuance of the bushfire situation felt by the DMs. From Table 18, the optimal decision alternative changes with state \(Z_{i}\) as the assessment levels vary. Therefore, the dynamic evolution of the emergencies must be captured during the emergency decision-making process. This is the main benefit of using the proposed method in dynamic emergency decision making.

  2. (2)

    The method in Yu and Lai 2011 focuses on exact values to make a fair comparison, lower bound, average point, upper bound of the interval values, which are aggregated from the interval-valued fuzzy hesitant numbers, to compute the rankings. It is thus important to note that the DMs exercised bounded rationality even in the context of risk and uncertainty. However, in extreme emergency response situations such as large bushfires, some DMs, for one reason or another, may be guided by their psychological behavior. However, the method in Yu and Lai 2011 ignores the psychological behavior of the DMs. What we have shown through this example is that ignoring the behavior of the DMs (e.g., sudden and untimely death of the firefighters during an emergency response) can lead to the wrong ranking results and hence an inferior final decision selection.

From the analysis, the proposed approach can offer several benefits, namely:

  1. (1)

    By using an IVHF set, the proposed method can better reflect the uncertainty of the DMs on the evaluation information. This is especially helpful for emergency decision-making problems characterized by incomplete information, uncertainty, and limited expertise.

  2. (2)

    The proposed method takes full account of the dynamics of the emergency events; the optimal choice can be adjusted dynamically based on the state. In this way, the DMs can exercise realistic judgement calls and improve the applicability of the proposed method.

  3. (3)

    The IVHF-TODIM method accommodates the DM’s psychological behavior in the context of risk and uncertainty for a large-scale emergency response situation and hence is better calibrated to accommodate the qualitative attributes of the DMs.

5 Conclusion

The emergency decision selection problem is important to emergency management, as the decisions made at critical moments can directly influence the performance of the emergency response. From this perspective, the proposed method offers much value. Many MADM methods have been used for emergency alternative selection problems, but they often neglect the DM’s psychological behavior. For that, the IVHF-TODIM method solves the dynamic emergency response problem by considering the DM’s psychological behavior. Compared to prospect theory which also considers the psychological behavior of the DM, the proposed method does not need to predetermine the aspiration levels of the attributes.

In this paper, the IVHF-TODIM method is introduced to handle the EDM problems under an IVHF environment, in which all attribute information provided to the DMs is in the form of IVHF sets. A new function to compute the degrees of gain and loss based on the geometric area method is used. Applying the probability density function and geometric area method, the gain and loss matrices of the interval belief degrees are developed. Next, we put forward a new expression to form the dominance degree matrix. The validity of the proposed method has been demonstrated using a recent example of an emergency alternative selection of a large bushfire in Australia during the start of summer. The novelty and value of the proposed method are shown through a comparative analysis.

Naturally, the proposed method has its limitations too. First, the weight of attributes is known in advance. Next, the proposed method only considers the attribute values to be in the form of IVHF sets. In some complex EDM problems, the attribute weights may be unknown or partially known, and the format of the attribute values may not be a single expression. Thus, for future work, one research direction is to supply a rigorous and realistic method for obtaining the attribute weights. We can also combine the proposed the interval-valued hesitant fuzzy TODIM method with other MADM techniques to deal with MADM problems posed under extreme uncertainty such as COVID-19.