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An innovative interval type-2 fuzzy approach for multi-scenario multi-project cash flow evaluation considering TODIM and critical chain with an application to energy sector

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Abstract

Project management has been proven to be an effective tool to manage sophisticated activities. Various techniques of project management have played an essential role for successful project implementation in different areas. Managing projects, especially in a multi-project environment, involves a complex situation because of its distinguishing feature in which a number of projects are being executed simultaneously, that is, they are followed in parallel. Therefore, applying human resources will be more effective and more idle time will be eliminated as well. More so, it can enable people to share their lessons learned from one project to another. With respect to handling a number of projects at the same time by most firms, it is sophisticated for contractors to cope with financial issues of projects, which involve different project cash inflows and outflows. Thus, taking an accurate cash flow prediction into account for projects has been changed into a crucial matter for firms, and lack of this consideration may result in the failure of projects as well. Moreover, there is a desperate need for uncertainties to be addressed thoroughly regarding their vital role for suitable project management. With these in mind, an innovative approach is proposed in this paper to anticipate the cash flow of project by considering type-2 fuzzy extension of both critical chain project management (CCM) for project scheduling and TODIM (an acronym in Portuguese for interactive Multi-Criteria Decision Making) method for selecting the best scenario in a multi-project environment. Hence, type-2 fuzzy numbers are utilized in order to state uncertainties. Eventually, a real-world project in a petro-refinery firm is applied to indicate the capability of the presented approach.

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Acknowledgements

The authors would like to thank the editor and anonymous referees for their valuable comments on the initial version of this study for the improvements.

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Authors and Affiliations

  1. Department of Industrial Engineering, Shahed University, Tehran, Iran

    Seyed Ali Mirnezami, Seyed Meysam Mousavi & Vahid Mohagheghi

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  1. Seyed Ali Mirnezami
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Appendix

Appendix

T2FSs that first were introduced by Zadeh [70] are development of type-1 fuzzy sets. These sets have turned out to be more appropriate compared with T1FSs in light of their importance to cope with uncertainties [6,7,8, 16, 29]. \( \tilde{\tilde{A}} \) can be denoted as follows [49]:

$$ \tilde{\tilde{A}} = \int_{x \in X} \int_{{u \in J_{X} }} \mu_{{\tilde{\tilde{A}}}} \left( {x, u} \right)/\left( {x, u} \right) $$
(63)

where \( J_{X} \subseteq \left[ {0,\; 1} \right] \) and \( \int \int \) represents union over all admissible \( x \) and \( u \).

An IT2FS \( \tilde{\tilde{A}} \) that is taken into account as a specific kind of T2FS, is shown as [49]:

$$ \tilde{\tilde{A}} = \int_{x \in X} \int_{{u \in J_{X} }} 1/\left( {x, u} \right) $$
(64)

where \( J_{X} \subseteq \left[ {0, \;1} \right] \).

TIT2FN is a particular type of IT2FN, whose upper membership function (UMF) and lower membership function (LMF) are determined as trapezoidal fuzzy numbers. These numbers are indicated as [14]:

$$ \begin{aligned} A = \left( {A^{\text{U}} , A^{\text{L}} } \right) = \left( {\left( {a_{1}^{\text{U}} , a_{2}^{\text{U}} , a_{3}^{\text{U}} , a_{4}^{\text{U}} ; H_{1} \left( {A^{\text{U}} } \right), H_{2} \left( {A^{\text{U}} } \right)} \right),} \right. \hfill \\ \left. {\left( {a_{1}^{\text{L}} , a_{2}^{\text{L}} , a_{3}^{\text{L}} , a_{4}^{\text{L}} ; H_{1} \left( {A^{\text{L}} } \right), H_{2} \left( {A^{\text{L}} } \right)} \right)} \right) \hfill \\ \end{aligned} $$
(65)

where \( A^{\text{U}} \) and \( A^{\text{L}} \) imply the UMF and LMF of \( A \), and \( H_{j} \left( {A^{\text{U}} } \right) \) and \( H_{j} \left( {A^{\text{L}} } \right) \) \( H_{j} \left( {A^{\text{U}} } \right) \in \left[ {0,\; 1} \right], H_{j} \left( {A^{\text{L}} } \right) \in \left[ {0, \;1} \right], \;j = 1, \;2 \) express the values of membership of the related factors \( a_{j + 1}^{\text{L}} \) and \( a_{j + 1}^{\text{U}} \), respectively. The membership function of a TIT2FS can be illustrated in Fig. 6.

Fig. 6
figure 6

Membership function of a trapezoidal IT2FS [14]

Full size image

If \( D_{1} \) and \( D_{2} \) are considered as two trapezoidal interval type-2 fuzzy numbers then:

$$ \begin{aligned} D_{1} = \left( {D_{1}^{\text{U}} , D_{1}^{\text{L}} } \right) = \left( {\left( {d_{11}^{\text{U}} , d_{12}^{\text{U}} , d_{13}^{\text{U}} , d_{14}^{\text{U}} ; H_{1} \left( {D_{1}^{\text{U}} } \right), H_{2} \left( {D_{1}^{\text{U}} } \right)} \right),} \right. \hfill \\ \left. {\left( {d_{11}^{\text{L}} , d_{12}^{\text{L}} , d_{13}^{\text{L}} , d_{14}^{\text{L}} ; H_{1} \left( {D_{1}^{\text{L}} } \right), H_{2} \left( {D_{1}^{\text{L}} } \right)} \right)} \right) \hfill \\ \end{aligned} $$
(66)
$$ \begin{aligned} D_{2} = \left( {D_{2}^{\text{U}} , D_{2}^{\text{L}} } \right) = \left( {\left( {d_{21}^{\text{U}} , d_{22}^{\text{U}} , d_{23}^{\text{U}} , d_{24}^{\text{U}} ; H_{1} \left( {D_{2}^{\text{U}} } \right), H_{2} \left( {D_{2}^{\text{U}} } \right)} \right),} \right. \hfill \\ \left. {\left( {d_{21}^{\text{L}} , d_{22}^{\text{L}} , d_{23}^{\text{L}} , d_{24}^{\text{L}} ; H_{1} \left( {D_{2}^{\text{L}} } \right), H_{2} \left( {D_{2}^{\text{L}} } \right)} \right)} \right) \hfill \\ \end{aligned} $$
(67)

The algebraic operations of them are based on [36] and can be determined as:

The addition operation between the aforementioned numbers can be defined as [36]:

$$ \begin{aligned} \widetilde{{\widetilde{D}}}_{1} \oplus \widetilde{{\widetilde{D}}}_{2} & = \left( {\widetilde{D}_{1}^{\text{U}} , \widetilde{D}_{1}^{\text{L}} } \right) + \left( {\widetilde{D}_{2}^{\text{U}} , \widetilde{D}_{2}^{\text{L}} } \right) \\ & = \left( {\left( {d_{11}^{\text{U}} + d_{21}^{\text{U}} , d_{12}^{\text{U}} + d_{22}^{\text{U}} , d_{13}^{\text{U}} + d_{23}^{\text{U}} , d_{14}^{\text{U}} + d_{24}^{\text{U}} ;} \right.} \right.\left( {\left( {H_{1} \left( {\widetilde{D}_{1}^{\text{U}} } \right)} \right.} \right. \\ & \quad \left. { + \left( {H_{1} \left( {\widetilde{D}_{2}^{\text{U}} } \right)} \right) - H_{1} \left( {\widetilde{D}_{1}^{\text{U}} } \right)* H_{1} \left( {\widetilde{D}_{2}^{\text{U}} } \right), \left( {H_{2} \left( {\widetilde{D}_{1}^{\text{U}} } \right)} \right) + H_{2} \left( {\widetilde{D}_{2}^{\text{U}} } \right)} \right) - H_{2} \left( {\widetilde{D}_{1}^{\text{U}} } \right) \\ & \quad \left. {*H_{2} \left( {\widetilde{D}_{2}^{\text{U}} } \right)} \right), d_{11}^{\text{L}} + d_{21}^{\text{L}} , d_{12}^{\text{L}} + d_{22}^{\text{L}} , d_{13}^{\text{L}} + d_{23}^{\text{L}} , d_{14}^{\text{L}} + d_{24}^{\text{L}} ;\left( {H_{1} \left( {\widetilde{D}_{1}^{\text{L}} } \right)} \right) \\ & \quad \left. {\left. { + H_{1} \left( {\widetilde{D}_{2}^{\text{L}} } \right) - H_{1} \left( {\widetilde{D}_{1}^{\text{L}} } \right)* H_{1} \left( {\widetilde{D}_{2}^{\text{L}} } \right), \left( {H_{2} \left( {\widetilde{D}_{1}^{\text{L}} } \right)} \right) + H_{2} \left( {\widetilde{D}_{2}^{\text{L}} } \right)) - H_{2} \left( {\widetilde{D}_{1}^{\text{L}} } \right)*H_{2} \left( {\widetilde{D}_{2}^{\text{L}} } \right)} \right)} \right). \\ \end{aligned} $$
(68)

The subtraction operation can be defined as [36]:

$$ \begin{aligned} \widetilde{{\widetilde{D}}}_{1} { \ominus } \widetilde{{\widetilde{D}}}_{2} & = \left( {\widetilde{D}_{1}^{\text{U}} , \widetilde{D}_{1}^{\text{L}} } \right){ \ominus }\left( {\widetilde{D}_{2}^{\text{U}} , \widetilde{D}_{2}^{\text{L}} } \right) \\ & = \left( {\left( {d_{11}^{\text{U}} - d_{24}^{\text{U}} , d_{12}^{\text{U}} - d_{23}^{\text{U}} , d_{13}^{\text{U}} - d_{22}^{\text{U}} , d_{14}^{\text{U}} - d_{21}^{\text{U}} ;} \right.} \right.\left( {\left( {H_{1} \left( {\widetilde{D}_{1}^{\text{U}} } \right)} \right.} \right. \\ & \quad \left. { + \left( {H_{1} \left( {\widetilde{D}_{2}^{\text{U}} } \right)} \right) - H_{1} \left( {\widetilde{D}_{1}^{\text{U}} } \right)* H_{1} \left( {\widetilde{D}_{2}^{\text{U}} } \right), \left( {H_{2} \left( {\widetilde{D}_{1}^{\text{U}} } \right)} \right) + H_{2} \left( {\widetilde{D}_{2}^{\text{U}} } \right)} \right) - H_{2} \left( {\widetilde{D}_{1}^{\text{U}} } \right) \\ & \quad \left. {* H_{2} \left( {\widetilde{D}_{2}^{\text{U}} } \right)} \right), d_{11}^{\text{L}} - d_{24}^{\text{L}} , d_{12}^{\text{L}} - d_{23}^{\text{L}} , d_{13}^{\text{L}} - d_{22}^{\text{L}} , d_{14}^{\text{L}} - d_{21}^{\text{L}} ;\left( {H_{1} \left( {\widetilde{D}_{1}^{\text{L}} } \right)} \right) \\ & \quad \left. {\left. {\left. {\left. { + H_{1} \left( {\widetilde{D}_{2}^{\text{L}} } \right)} \right) - H_{1} \left( {\widetilde{D}_{1}^{\text{L}} } \right)* H_{1} \left( {\widetilde{D}_{2}^{\text{L}} } \right), \left( {H_{2} \left( {\widetilde{D}_{1}^{\text{L}} } \right)} \right) + H_{2} \left( {\widetilde{D}_{2}^{\text{L}} } \right)} \right) - H_{2} \left( {\widetilde{D}_{1}^{\text{L}} } \right)* H_{2} \left( {\widetilde{D}_{2}^{\text{L}} } \right)} \right)} \right). \\ \end{aligned} $$
(69)

The multiplication operation can be defined as [36]:

$$ \begin{aligned} \widetilde{{\widetilde{D}}}_{1} \otimes \widetilde{{\widetilde{D}}}_{2} & = \left( {\widetilde{D}_{1}^{\text{U}} , \widetilde{D}_{1}^{\text{L}} } \right) \otimes \left( {\widetilde{D}_{2}^{\text{U}} , \widetilde{D}_{2}^{\text{L}} } \right) \\ & = \left( {\left( {Y_{11}^{\text{U}} , Y_{12}^{\text{U}} , Y_{13}^{\text{U}} , Y_{14}^{\text{U}} ;{ \hbox{min} }\left( {H_{1} \left( {\tilde{d}_{1}^{\text{U}} } \right), H_{1} \left( {\tilde{d}_{2}^{\text{U}} } \right)} \right),\hbox{min} \left( { H_{2} \left( {\tilde{d}_{1}^{\text{U}} } \right), H_{2} \left( {\tilde{d}_{2}^{\text{U}} } \right)} \right)} \right)} \right., \\ & \quad \left. {Y_{11}^{\text{L}} , Y_{12}^{\text{L}} , Y_{13}^{\text{L}} , Y_{14}^{\text{L}} ;\hbox{min} \left( {H_{1} \left( {\tilde{d}_{1}^{\text{L}} } \right), H_{1} \left( {\tilde{d}_{2}^{\text{L}} } \right)} \right), \hbox{min} \left( {H_{2} \left( {\tilde{d}_{1}^{\text{L}} } \right), H_{2} \left( {\tilde{d}_{2}^{\text{L}} } \right)} \right)} \right). \\ {\text{where}}\;Y_{i}^{T} & = \left\{ {\begin{array}{*{20}l} {\hbox{min} \left( {d_{1i}^{T} d_{2i}^{T} , d_{1i}^{T} d_{{2\left( {5 - i} \right)}}^{T} , d_{{1\left( {5 - i} \right)}}^{T} d_{2i}^{T} , d_{{1\left( {5 - i} \right)}}^{T} d_{{2\left( {5 - i} \right)}}^{T} } \right) } \hfill & {{\text{if}}\;i = 1, \;2\quad T \in \left\{ {U, L} \right\}} \hfill \\ {\hbox{max} \left( {d_{1i}^{T} d_{2i}^{T} , d_{1i}^{T} d_{{2\left( {5 - i} \right)}}^{T} , d_{{1\left( {5 - i} \right)}}^{T} d_{2i}^{T} , d_{{1\left( {5 - i} \right)}}^{T} d_{{2\left( {5 - i} \right)}}^{T} } \right) } \hfill & {{\text{if}}\;i = 3, \;4 \quad T \in \left\{ {U, L} \right\}} \hfill \\ \end{array} } \right. \\ \end{aligned} $$
(70)

The above-mentioned operation between a crisp value \( \beta \) and \( \widetilde{{\widetilde{D}}}_{1} \) can be defined as [36]:

$$ \begin{aligned} \beta \widetilde{{\widetilde{D}}}_{1} & = \left[ {\left( {\left( {\beta d_{11}^{\text{U}} ,\beta d_{12}^{\text{U}} ,\beta d_{13}^{\text{U}} , \beta d_{14}^{\text{U}} ;1 - \left( {1 - H_{1} \left( {\tilde{d}_{1}^{\text{U}} } \right)} \right)^{\beta } , 1 - \left( {1 - H_{2} \left( {\tilde{d}_{1}^{\text{U}} } \right)} \right)^{\beta } } \right),} \right.} \right. \\ & \quad \left. {\left( {\beta d_{11}^{\text{L}} ,\beta d_{12}^{\text{L}} ,\beta d_{13}^{\text{L}} , \beta d_{14}^{\text{L}} ; 1 - \left( {1 - H_{1} \left( {\tilde{d}_{1}^{\text{L}} } \right)} \right)^{\beta } ,\left( {1 - \left( {1 - H_{2} \left( {\tilde{d}_{1}^{\text{L}} } \right)} \right)^{\beta } } \right)} \right)} \right] \\ \end{aligned} $$
(71)

and

$$ \begin{aligned} \widetilde{{\widetilde{D}}}_{1}^{\beta } & = \left[ {\left( {\left( {d_{11}^{\text{U}} } \right)^{\beta } , \left( {d_{12}^{\text{U}} } \right)^{\beta } , \left( {d_{13}^{\text{U}} } \right)^{\beta } , \left( {d_{14}^{\text{U}} } \right)^{\beta } ;\left( {H_{1} \left( {\tilde{d}_{1}^{\text{U}} } \right)} \right)^{\beta } , \left( {H_{2} \left( {\tilde{d}_{1}^{\text{U}} } \right)} \right)^{\beta } } \right),} \right. \\ & \quad \left. {\left( { \left( {d_{11}^{\text{L}} } \right)^{\beta } , \left( {d_{12}^{\text{L}} } \right)^{\beta } , \left( {d_{13}^{\text{L}} } \right)^{\beta } , \left( {d_{14}^{\text{L}} } \right)^{\beta } ; \left( {H_{1} \left( {\tilde{d}_{1}^{\text{L}} } \right)} \right)^{\beta } , \left( {H_{2} \left( {\tilde{d}_{1}^{\text{L}} } \right)} \right)^{\beta } } \right)} \right] \\ \end{aligned} $$
(72)

The division operation can be defined as follows [25]:

$$ \begin{aligned} D_{1} { \oslash } D_{2} & = \left( {D_{1}^{\text{U}} , D_{1}^{\text{L}} } \right){ \oslash } \left( {D_{2}^{\text{U}} , D_{2}^{\text{L}} } \right) \\ & = \left( {\left( {Y_{11}^{\text{U}} , Y_{12}^{\text{U}} , Y_{13}^{\text{U}} , Y_{14}^{\text{U}} ;\hbox{min} \left( {H_{1} \left( {D_{1}^{\text{U}} } \right), H_{1} \left( {D_{2}^{\text{U}} } \right)} \right),\hbox{min} \left( {H_{2} \left( {D_{1}^{\text{U}} } \right), H_{2} \left( {D_{2}^{\text{U}} } \right)} \right)} \right),} \right. \\ & \quad \left. {\left( {Y_{11}^{\text{L}} , Y_{12}^{\text{L}} , Y_{13}^{\text{L}} , Y_{14}^{\text{L}} ;\hbox{min} \left( {H_{1} \left( {D_{1}^{\text{L}} } \right), H_{1} \left( {D_{2}^{\text{L}} } \right)} \right), { \hbox{min} } \left( {H_{2} \left( {D_{1}^{\text{L}} } \right), H_{2} \left( {D_{2}^{\text{L}} } \right)} \right)} \right)} \right) \\ & \quad {\text{where}}\;Y_{1i}^{T} = \hbox{min} \left( {\frac{{d_{1i}^{T} }}{{d_{2i}^{T} }},\frac{{d_{1i}^{T} }}{{d_{{2\left( {5 - i} \right)}}^{T} }},\frac{{d_{{1\left( {5 - i} \right)}}^{T} }}{{d_{2i}^{T} }},\frac{{d_{{1\left( {5 - i} \right)}}^{T} }}{{d_{{2\left( {5 - i} \right)}}^{T} }}} \right), \quad T \in \left\{ {U, L} \right\},\quad i \in \left\{ {1, 2} \right\} \\ x_{1j}^{T} & = { \hbox{max} }\left( {\frac{{d_{{1\left( {5 - j} \right)}}^{T} }}{{d_{{2\left( {5 - j} \right)}}^{T} }},\frac{{d_{{1\left( {5 - j} \right)}}^{T} }}{{d_{2j}^{T} }},\frac{{d_{1j}^{T} }}{{d_{{2\left( {5 - j} \right)}}^{T} }},\frac{{d_{1j}^{T} }}{{d_{2j}^{T} }}} \right), \quad T \in \left\{ {U, L} \right\}, \quad j \in \left\{ {3, 4} \right\} \\ \end{aligned} $$
(73)

Euclidean distance between them can be defined as [74]:

$$ d\left( {D_{1} , D_{2} } \right) = \sqrt {\mathop \sum \limits_{b = 1}^{4} \left( {d_{1b}^{\text{U}} - d_{2b}^{\text{U}} } \right)^{2} + \mathop \sum \limits_{b = 1}^{4} \left( {d_{1b}^{\text{L}} - d_{2b}^{\text{L}} } \right)^{2} + \mathop \sum \limits_{b = 1}^{2} \left( {H_{b} \left( {D_{1}^{\text{U}} } \right) - H_{b} \left( {D_{2}^{\text{U}} } \right)} \right)^{2} + \mathop \sum \limits_{b = 1}^{2} \left( {H_{b} \left( {D_{1}^{\text{L}} } \right) - H_{b} \left( {D_{2}^{\text{L}} } \right)} \right)^{2} } $$
(74)

where \( b \) indicates number of components upper, lower, and membership degree of trapezoidal interval type-2 fuzzy number.

The expected value of \( D \) as a trapezoidal interval type-2 fuzzy number can be defined as follows [36]:

$$ E\left( D \right) = \frac{1}{2}\left( {\frac{1}{4}\mathop \sum \limits_{i = 1}^{4} \left( {d_{i}^{\text{L}} + d_{i}^{\text{U}} } \right)} \right) \times \frac{1}{4}\left( {\mathop \sum \limits_{i = 1}^{2} \left( {H_{i} \left( {D^{\text{L}} } \right) + H_{i} \left( {D^{\text{U}} } \right)} \right)} \right) $$
(75)

It is believed that by taking \( D_{1} \) and \( D_{2} \) into account as a trapezoidal interval type-2 fuzzy number, if and only if \( E\left( {D_{1} } \right) > E\left( {D_{2} } \right) \) then \( D_{1} > D_{2} \)

\( D_{s} \) and \( D_{t} \) are considered to be as two trapezoidal IT2FN below [36]:

$$ \begin{aligned} D_{s} = \left( {D_{s}^{\text{U}} , D_{s}^{\text{L}} } \right) = \left( {\left( {d_{s1}^{\text{U}} , d_{s2}^{\text{U}} , d_{s3}^{\text{U}} , d_{s4}^{\text{U}} ; H_{1} \left( {D_{s}^{\text{U}} } \right), H_{2} \left( {D_{s}^{\text{U}} } \right)} \right),} \right. \hfill \\ \left. {\left( {d_{s1}^{\text{L}} , d_{s2}^{\text{L}} , d_{s3}^{\text{L}} , d_{s4}^{\text{L}} ; H_{1} \left( {D_{s}^{\text{L}} } \right), H_{2} \left( {D_{s}^{\text{L}} } \right)} \right)} \right) \hfill \\ \end{aligned} $$
(76)

and

$$ \begin{aligned} D_{t} = \left( {D_{t}^{\text{U}} , D_{t}^{\text{L}} } \right) = \left( {\left( {d_{t1}^{\text{U}} , d_{t2}^{\text{U}} , d_{t3}^{\text{U}} , d_{t4}^{\text{U}} ; H_{1} \left( {D_{t}^{\text{U}} } \right), H_{2} \left( {D_{t}^{\text{U}} } \right)} \right),} \right. \hfill \\ \left. {\left( {d_{t1}^{\text{L}} , d_{t2}^{\text{L}} , d_{t3}^{\text{L}} , d_{t4}^{\text{L}} ; H_{1} \left( {D_{t}^{\text{L}} } \right), H_{2} \left( {D_{t}^{\text{L}} } \right)} \right)} \right) \hfill \\ \end{aligned} $$
(77)

Let

$$ Y = \frac{{\mathop \sum \nolimits_{{T \in \left\{ {L, U} \right\}}} \left( {\left( {d_{s3}^{T} + d_{s4}^{T} } \right) - \left( {d_{t1}^{T} + d_{t2}^{T} } \right)} \right) + \mathop \sum \nolimits_{k = 1}^{2} \left( {\hbox{max} \left( {H_{k} \left( {D_{s}^{\text{U}} } \right) - H_{k} \left( {D_{t}^{\text{U}} } \right), 0} \right) + \hbox{max} \left( {H_{k} \left( {D_{s}^{\text{L}} } \right) - H_{k} \left( {D_{t}^{\text{L}} } \right), 0} \right)} \right)}}{{\mathop \sum \nolimits_{k = 1}^{4} {\text{len}} \left( {v_{k} } \right) + \mathop \sum \nolimits_{k = 1}^{2} \left| {H_{k} \left( {D_{s}^{\text{U}} } \right) - H_{k} \left( {D_{t}^{\text{U}} } \right)} \right| + \mathop \sum \nolimits_{k = 1}^{2} \left| {H_{k} \left( {D_{s}^{\text{L}} } \right) - H_{k} \left( {D_{t}^{\text{L}} } \right)} \right|}} $$
(78)

where \( {\text{len}} \left( {v_{1} } \right) = d_{s4}^{\text{L}} + d_{s3}^{\text{L}} - d_{s1}^{\text{L}} - d_{s2}^{\text{L}} , \) \( {\text{len}} \left( {v_{2} } \right) = d_{s4}^{\text{U}} + d_{s3}^{\text{U}} - d_{s1}^{\text{U}} - d_{s2}^{\text{U}} , \) \( {\text{len}} \left( {v_{3} } \right) = d_{t4}^{\text{L}} + d_{t3}^{\text{L}} - d_{t1}^{\text{L}} - d_{t2}^{\text{L}} , \) \( {\text{len}} \left( {v_{4} } \right) = d_{t4}^{\text{U}} + d_{t3}^{\text{U}} - d_{t1}^{\text{U}} - d_{t2}^{\text{U}} \), respectively.

Thus, the possibility degree of \( D_{s} \) over \( D_{t} \) will be as follows [36]:

$$ p\left( {D_{s} \ge D_{t} } \right) = { \hbox{min} }\left( {\hbox{max} \left( {Y,0} \right),1} \right) $$
(79)

In accordance with the above-mentioned equations, if \( p\left( {D_{s} \ge D_{t} } \right) < 0.5 \), then \( D_{s} < D_{t} \).

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Mirnezami, S.A., Mousavi, S.M. & Mohagheghi, V. An innovative interval type-2 fuzzy approach for multi-scenario multi-project cash flow evaluation considering TODIM and critical chain with an application to energy sector. Neural Comput & Applic 33, 2263–2284 (2021). https://doi.org/10.1007/s00521-020-05095-z

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