Multi-item two-stage fixed-charge 4DTP with hybrid random type-2 fuzzy variable

Abstract

Parameters of some real-life decision-making problems are simultaneously uncertain, imprecise, and vague. In this paper, for the first time, we introduce two such new hybrid uncertain variables—random type-2 trapezoidal (RT2TF) and gamma fuzzy (RT2GF) variables, their derandomization and defuzzification methods, and applications. Mimicking two-stage public distribution system of the developing countries, breakable multi-item two-stage fixed-charge four-dimensional transportation problems (MITSFC-4DTPs) are formulated and solved. Here, some breakable items are transported from sources to destinations via warehouses using some conveyances, traveling through connecting routes and incurring transportation costs and fixed charges at each stage. The objective is to find suitable conveyances, appropriate travel routes, and corresponding transported amounts at each stage so that total transportation cost is minimum. The model’s parameters—transportation costs, fixed costs, availabilities, demands, and conveyances’ capacities are considered as RT2TF and RT2GF. The models’ random type-2 fuzzy objectives and constraints are first derandomized using expectation and probability chance constraint techniques, respectively. The reduced type-2 fuzzy models are transformed into type-1 fuzzy problems by the CV-based reduction technique (CV-bRT), which are then converted to deterministic ones using two methods—generalized credibility measures (GCM) theory and centroid techniques (trapezoidal fuzzy problem only) separately. All these deterministic models are solved by the generalized reduced gradient (GRG) method using LINGO 12.0 and numerically illustrated. A real-life problem and several particular models under different uncertain environments are solved using some input data. Results from two CV-based methods—CV-bRT-GCM and CV-bRT-centroid for type-2 fuzzy, are compared, and superiority of proposed CV-bRT-GCM is established. In 4DTPs, the importance of multi-routes is numerically illustrated. Some managerial insights are also presented.

Appendices

Background and preliminaries

1.1 Random variable

A random variable ${\hat{X}}$ is a mapping from the probability space (S, $\varOmega$, Pr) to the set of real numbers $\mathbb {R}$ where S is a sample space corresponding to a random experiment E and $\varOmega$ be the non-empty set of all events (i.e., subsets of S) which is closed under arbitrary union and complements in S and Pr is the probability function from $\varOmega$ to [0,1] such that $Pr(\{x_i \})=p_i,~ 0\le p_i \le 1, ~ \forall \, x_i \in S(i=1, 2, ...)$ and $\sum \nolimits _{i=1}^{\infty }p_i=1$.

1.2 Type-1 (T1FV) and type-2 fuzzy variable (T2FV)

Possibility Space. A triplet $(\theta , P(\theta ), Pos)$ is called possibility space, if it satisfies the following condition. (i) $P(\theta )=1, P(\phi )=0$ (ii) for any $A_i | i \in I \subset P(\theta ), Pos \{ \bigcup \nolimits _{i \in I}A_i \}= \sup \nolimits _{i \in I} Pos \{ A_i\}$ where $\theta$ be non-empty set, $\phi$ be the null set, $P(\theta )$ be the power set of $P(\theta )$, $Pos:P(\theta )\rightarrow [0,1]$.

A T1FV ${\tilde{\varsigma }}$ is a mapping from the possibility space $(\theta , P(\theta ), Pos)$ to the real numbers $\mathbb {R}$.

The possibility measure (Pos) of fuzzy event $\{ {\tilde{\varsigma }} \in B \}$ is defined as $Pos\{ {\tilde{\varsigma }} \in B \}=\sup \nolimits _{x \in B} \mu _{{\tilde{\varsigma }}} (x)$. $\mu _{{\tilde{\varsigma }}} (x)$ represents the possibility distribution of ${\tilde{\varsigma }}$.

The necessity measure (Nec) is defined as $Nes\{ {\tilde{\varsigma }} \in B \}=1-Pos\{ {\tilde{\varsigma }} \in B^c \}=1-\sup \nolimits _{x \in B^c} \mu _{{\tilde{\varsigma }}} (x)$.

The credibility measure is defined as $Cr\{ {\tilde{\varsigma }} \in B \}=\frac{1}{2}(Pos\{ {\tilde{\varsigma }} \in B \}+Nes\{ {\tilde{\varsigma }} \in B \})$.

The generalized credibility measure ${\tilde{Cr}}$ of the event $\{{\tilde{\varsigma }} \in B \}$is defined as $Cr\{ {\tilde{\varsigma }} \in B \}=\frac{1}{2}(\sup \nolimits _{x \in \mathbb {R}}\mu _{{\tilde{\varsigma }}} (x)+\sup \nolimits _{x \in B}\mu _{{\tilde{\varsigma }}} (x)-\sup \nolimits _{x \in B^{c}}\mu _{{\tilde{\varsigma }}} (x) )$, where ${\tilde{\varsigma }}$ is generalized fuzzy variable with the distribution $\mu$. If ${\tilde{\varsigma }}$ is normalized, it is easy to verify that $Cr({\tilde{\varsigma }}\in B)+Cr({\tilde{\varsigma }}\in B^{c})=\sup \nolimits _{x\in \mathbb {R}}\mu _{{\tilde{\varsigma }}} (x) =1$ and then ${\tilde{Cr}}$ coincides with the usual credibility measure.

Regular fuzzy variable: If a fuzzy variable with the membership function $\mu _{{\tilde{\varsigma }}} (x)$ satisfied $\sup \nolimits _{x \in \mathbb {R}} \mu _{{\tilde{\varsigma }}} (x)=1$, we call this fuzzy variable as regular fuzzy variable (RFV).

Fuzzy possibility space: (Liu and Liu 2007) A triplet $(\theta , P(\theta ), {\tilde{P}}os)$ is referred to as a fuzzy possibility space (FPS), if it satisfies the following conditions:

• (i) ${\tilde{P}}os\{\phi \} ={\tilde{0}}$,

• (ii) for any subclass ${A_i|i \in I}$ of $P(\theta )$ (finite, countable or uncountable), ${\tilde{P}}os \left\{ \bigcup \nolimits _{i \in I}A_i \right\} = \sup \nolimits _{i \in I} {\tilde{P}}os \{A_i\}$, where $P(\theta )$ be an ample field on the universe $\theta , F([0, 1])$ the set of all regular fuzzy variables on [0,1],and ${\tilde{P}}os : P(\theta )\rightarrow F([0, 1])$ a set function on $P(\theta )$ such that ${{\tilde{P}}os(A)|atom A \in P(\theta )}$ is a family of mutually independent RFVs.

Moreover, if $\mu _{{\tilde{P}}os(\theta )}(1) = 1$, then we call ${\tilde{Pos}}$ a regular fuzzy possibility measure.

A T2FV is a mapping from the fuzzy possibility space $(\theta , P(\theta ),{\tilde{P}}os)$ to real numbers $\mathbb {R}$.

1.3 Type-1 regular fuzzy variable

A type-1 trapezoidal fuzzy variable is denoted by ${\tilde{\varsigma }}=(a_1, a_2, a_3, a_4)$.

A type-1 gamma fuzzy variable (Liu and Liu 2010) is denoted by ${\tilde{\varsigma }}={\tilde{\gamma }} (\lambda , r)$. Then, the possibility distribution of $\varsigma$ is defined as, $\mu _\varsigma (u)={(\frac{u}{\lambda r})}^r exp(r-\frac{u}{\lambda }), ~ u\in [0,1]$ where the parameter $0 < r \le 1$ and $0 < \lambda \le \frac{1}{r}$.

1.4 Type-2 regular fuzzy variable

A type-2 trapezoidal fuzzy variable is denoted by ${\tilde{\varsigma }}=(a_1, a_2, a_3, a_4; \theta _l, \theta _r)$. The secondary possibility distribution is also denoted by ${\tilde{\mu }}_{{\tilde{\varsigma }}}(x)$. It is defined as a type-1 regular trapezoidal fuzzy variable (Qin et al. 2011), i.e.,

$$\begin{aligned}&{\tilde{\mu }}_{{\tilde{\varsigma }}}(x)=\left( \frac{x-a_1}{a_2-a_1}-\theta _l \text {min} \left\{ \frac{x-a_1}{a_2-a_1}.\frac{a_2-x}{a_2-a_1} \right\} , \right. \\&\quad \left. \frac{x-a_1}{a_2-a_1}, \frac{x-a_1}{a_2-a_1}+\theta _r \text {min} \left\{ \frac{x-a_1}{a_2-a_1}.\frac{a_2-x}{a_2-a_1} \right\} \right) \end{aligned}$$

for any $x \in [a_1, a_2]$, and for any $x \in (a_2, a_3]$, the secondary possibility distribution is 1, and

$$\begin{aligned}&{\tilde{\mu }}_{{\tilde{\varsigma }}}(x)=\left( \frac{a_4-x}{a_4-a_3}-\theta _l \text {min} \left\{ \frac{a_4-x}{a_4-a_3}.\frac{x-a_3}{a_4-a_3} \right\} , \frac{a_4-x}{a_4-a_3},\right. \\&\quad \left. \frac{a_4-x}{a_4-a_3}+\theta _r \text {min} \left\{ \frac{a_4-x}{a_4-a_3}.\frac{x-a_3}{a_4-a_3} \right\} \right) \end{aligned}$$

for any $x \in (a_3, a_4]$, where $\theta _l, \theta _r \in [0, 1]$ are two parameters characterizing the degree of uncertainty of $\varsigma$ taking the value x.

If $a_2 =a_3$, type-2 trapezoidal fuzzy variable is reduced to type-2 triangular fuzzy variable. In similar way, we can write secondary possibility distribution of type-2 triangular fuzzy number.

A type-2 gamma fuzzy variable is denoted by ${\tilde{\varsigma }}={\tilde{\gamma }} (\lambda , r; \theta _l, \theta _r)$. The secondary possibility distribution is also denoted by ${\tilde{\mu }}_{{\tilde{\varsigma }}}(x)$. It is defined as a type-1 regular gamma fuzzy variable (Qin et al. 2011), i.e.,

$$\begin{aligned} {\tilde{\mu }}_{{\tilde{\varsigma }}}(x)=&({(\frac{x}{\lambda r})}^r exp(r-\frac{x}{\lambda })-\theta _l ~ \text {min} \left\{ 1-{(\frac{x}{\lambda r})}^r exp(r\right. \\&\left. -\frac{x}{\lambda }),{(\frac{x}{\lambda r})}^r exp(r-\frac{x}{\lambda }) \right\} , {(\frac{x}{\lambda r})}^r exp(r-\frac{x}{\lambda }),\\&{(\frac{x}{\lambda r})}^r exp(r-\frac{x}{\lambda })+\theta _r ~ \text {min}\left\{ 1-{(\frac{x}{\lambda r})}^r exp(r\right. \\&\left. -\frac{x}{\lambda }),{(\frac{x}{\lambda r})}^r exp(r-\frac{x}{\lambda })\right\} ) \end{aligned}$$

or any $x \in \mathbb {R}$, where $\lambda > 0$, r is a fixed constant, and $\theta _l, \theta _r \in [0, 1]$ are two parameters characterizing the degree of uncertainty of ${\tilde{\varsigma }}$ taking the value x.

1.5 Random type-2 regular fuzzy variable

Let $(U, \varOmega , {\tilde{Pos}})$ be a fuzzy possibility space. Then, random type-2 fuzzy variable is a measurable function from U to set of random variables.

For the type-2 trapezoidal fuzzy variable $(a_{1}, a_{2}, a_{3}, a_{4}; \theta _{l}, \theta _{r})$ if the parameters $a_1$, $a_2$, $a_3$, $a_4$, $\theta _l$, $\theta _r$ are all random then the variable is RT2TF variable. It is denoted by $({\hat{a}}_{1}, {\hat{a}}_2, {\hat{a}}_3, {\hat{a}}_4; {\hat{\theta }}_{l}, {\hat{\theta }}_{r})$.

If $a_{2}= a_{3}$ then type-2 trapezoidal fuzzy variable $(a_{1}, a_{2}, a_{3}, a_{4}; \theta _{l}, \theta _{r})$ becomes type-2 triangular fuzzy variable $(a_1, a_2, a_3; \theta _l, \theta _r)$ and the parameters $a_1, a_2, a_3, \theta _l, \theta _r$ are all random then the variable is random type-2 triangular fuzzy variable. It is denoted by $({\hat{a}}_{1}, {\hat{a}}_{2}, {\hat{a}}_{3}; {\hat{\theta }}_{l}, {\hat{\theta }}_{r})$.

For the type-2 gamma fuzzy variable ${\tilde{\gamma }} (\lambda , r; \theta _l, \theta _r)$ if the parameters ${\tilde{\gamma }} (\lambda , r; \theta _l, \theta _r)$ are all random then the variable is RT2GF variable. It is denoted by ${\tilde{\gamma }} ({\hat{\lambda }}, {\hat{r}}; \hat{\theta _{l}}, \hat{\theta _{r}})$.

1.6 Critical values (CV) for regular fuzzy variable

Here, three kinds of CV for regular fuzzy variable is defined using fuzzy integral.

• (i) The optimistic CV of $\varsigma$ is denoted by $CV^*[\varsigma ]$ and is given by $CV^*[\varsigma ] = \sup \nolimits _{\alpha \in [0,1]}[\alpha \wedge Pos(\varsigma \ge \alpha )]$

• (ii) The pessimistic CV of $\varsigma$ is denoted by $CV_*[\varsigma ]$ and is given by $CV_*[\varsigma ] = \sup \nolimits _{\alpha \in [0,1]}[\alpha \wedge Nes(\varsigma \ge \alpha )]$

• (iii) The CV of $\varsigma$ is denoted by $CV_*[\varsigma ]$ and is given by, $CV[\varsigma ] = \sup \nolimits _{\alpha \in [0,1]}[\alpha \wedge Cr(\varsigma \ge \alpha )]$.

Some theorems and corollaries

Theorem 1

(Qin et al. 2011) Let ${\tilde{\varsigma }}$ be a type-2 trapezoidal fuzzy variable defined as ${\tilde{\varsigma }} = (a_1, a_2, a_3, a_4; \theta _l, \theta _r )$. Then, we have:

(i) Using the optimistic CV reduction method, the reduction $\varsigma _1$ of ${\tilde{\varsigma }}$ has the following possibility distribution:

$$\begin{aligned} \mu _{\varsigma _1}(x)= {\left\{ \begin{array}{ll} \frac{(1+\theta _r)(x-a_1)}{a_2-a_1+\theta _r(x-a_1)}, &{}\text { if } x \in \left[ a_1,\frac{a_1+a_2}{2} \right] \\ \frac{(1-\theta _r)x+\theta _r a_2-a_1}{a_2-a_1+\theta _r(a_2-x)}, &{}\text { if } x \in \left( \frac{a_1+a_2}{2},a_2 \right] \\ 1, &{}\text { if } x \in \left( a_2,a_3 \right] \\ \frac{(-1+\theta _r)x-\theta _r a_3+a_4}{a_4-a_3+\theta _r(x-a_3)}, &{}\text { if } x \in \left( a_3,\frac{a_3+a_4}{2} \right] \\ \frac{(1+\theta _r)(a_4-x)}{a_4-a_3+\theta _r(a_4-x)}, &{}\text { if } x \in \left( \frac{a_3+a_4}{2},a_4 \right] \end{array}\right. } \end{aligned}$$

(ii) Using the pessimistic CV reduction method, the reduction $\varsigma _2$ of ${\tilde{\varsigma }}$ has the following possibility distribution:

$$\begin{aligned} \mu _{\varsigma _2}(x)= {\left\{ \begin{array}{ll} \frac{x-a_1}{a_2-a_1+\theta _l(x-a_1)}, &{}\text { if } x \in \left[ a_1,\frac{a_1+a_2}{2} \right] \\ \frac{x-a_1}{a_2-a_1+\theta _l(a_2-x)}, &{}\text { if } x \in \left( \frac{a_1+a_2}{2},a_2 \right] \\ 1, &{}\text { if } x \in \left( a_2,a_3 \right] \\ \frac{a_4-x}{a_4-a_3+\theta _l(x-a_3)}, &{}\text { if } x \in \left( a_3,\frac{a_3+a_4}{2} \right] \\ \frac{a_4-x}{a_4-a_3+\theta _l(a_4-x)}, &{}\text { if } x \in \left( \frac{a_3+a_4}{2},a_4 \right] \end{array}\right. } \end{aligned}$$

(iii) Using the CV reduction method, the reduction $\varsigma _3$ of ${\tilde{\varsigma }}$ has the following possibility distribution:

$$\begin{aligned} \mu _{\varsigma _3}(x)= {\left\{ \begin{array}{ll} \frac{(1+\theta _r)(x-a_1)}{a_2-a_1+2\theta _r(x-a_1)}, &{}\text { if } x \in \left[ a_1,\frac{a_1+a_2}{2} \right] \\ \frac{(1-\theta _l)x+\theta _l a_2-a_1}{a_2-a_1+2\theta _l(a_2-x)}, &{}\text { if } x \in \left( \frac{a_1+a_2}{2},a_2 \right] \\ 1, &{}\text { if } x \in \left( a_2,a_3 \right] \\ \frac{(-1+\theta _l)x-\theta _l a_3+a_4}{a_4-a_3+2\theta _l(x-a_3)}, &{}\text { if } x \in \left( a_3,\frac{a_3+a_4}{2} \right] \\ \frac{(1+\theta _r)(a_4-x)}{a_4-a_3+2\theta _r(a_4-x)}, &{}\text { if } x \in \left( \frac{a_3+a_4}{2},a_4 \right] \end{array}\right. } \end{aligned}$$

Figures 5,  6 and  7 represent the possibility distribution of $\varsigma _1$, $\varsigma _2$ and $\varsigma _3$, respectively, defined in Theorem 1.

Fig. 5

The possibility distributions of $\mu _{\varsigma (x)}$, a type-2 trapezoidal fuzzy number, using optimistic reduction method

Fig. 6

The possibility distributions of $\mu _{\varsigma (x)}$, a type-2 trapezoidal fuzzy number, using pessimistic reduction method

Fig. 7

The possibility distributions of $\mu _{\varsigma (x)}$, a type-2 trapezoidal fuzzy number, using CV reduction method

Theorem 2

Let $\varsigma _i$ be the reduction of the type-2 trapezoidal fuzzy variable $\tilde{\varsigma _i}$=($a^i_1$, $a^i_2$, $a^i_3$, $a^i_4$; $\theta _{l,i}$, $\theta _{r,i})$ obtained by the CV reduction method for $i = 1, 2,..., n$. Suppose $\varsigma _1, \varsigma _2,..., \varsigma _n$ are mutually independent, and $k_i \ge 0$ for $i = 1, 2,..., n$. The different level of $\alpha$ of generalized credibility, then ${\tilde{Cr}} \{ \sum \nolimits _{i=1}^n k_i \varsigma _i \le u \} \ge \alpha$ is equivalent to

$$\begin{aligned}&\sum ^n_{i=1} \frac{(1-2\alpha +(1-4\alpha )\theta _{r,i})k_i a^i_1+2\alpha k_i a^i_2}{1+(1-4\alpha ) \theta _{r,i}}\\&\le u \qquad \text {if}~ \alpha \in (0, 0.25]\\&\sum ^n_{i=1} \frac{(1-2\alpha )k_i a^i_1+(2\alpha + (4\alpha -1)\theta _{l,i}) k_i a^i_2}{1+(4\alpha -1) \theta _{l,i}}\\&\le u \qquad \text {if}~ \alpha \in (0.25, 0.5] \\&\sum ^n_{i=1} \frac{(2\alpha -1)k_i a^i_4+(2(1-\alpha ) + (3-4\alpha )\theta _{l,i}) k_i a^i_3}{1+(3-4\alpha ) \theta _{l,i}} \\&\le u \qquad \text {if}~ \alpha \in (0.5, 0.75]\\&\sum ^n_{i=1} \frac{(2\alpha -1+(4\alpha -3)\theta _{r,i})k_i a^i_4+2(1-\alpha ) k_i a^i_3}{1+(4\alpha -3) \theta _{r,i}}\\&\le u \qquad \text {if}~ \alpha \in (0.75, 1]. \end{aligned}$$

Proof

We only prove for $\alpha >0.5$. Other can be proved similarly. For each i= 1,2,...,n, since $\varsigma _i$ is the reduction of the type-2 trapezoidal fuzzy variable $\tilde{\varsigma _i}$ obtained by the CV reduction method, we know that the fuzzy variable $\varsigma _i$ has the following possibility distribution:

$$\begin{aligned} \mu _{\varsigma _i}(x)= {\left\{ \begin{array}{ll} \frac{(1+\theta _{r,i})(x-a^i_1)}{a^i_2-a^i_1+2\theta _{r,i}(x-a^i_1)}, &{}\text { if } x \in \left[ a^i_1,\frac{a^i_1+a^i_2}{2} \right] \\ \frac{(1-\theta _{l,i})x+\theta _{l,i} a^i_2-a^i_1}{a^i_2-a^i_1+2\theta _{l,i}(a^i_2-x)}, &{}\text { if } x \in \left( \frac{a^i_1+a^i_2}{2},a^i_2 \right] \\ 1, &{}\text { if } x \in \left( a^i_2,a^i_3 \right] \\ \frac{(-1+\theta _{l,i})x-\theta _{l,i} a^i_3+a^i_4}{a^i_4-a^i_3+2\theta _{l,i}(x-a^i_3)}, &{}\text { if } x \in \left( a^i_3,\frac{a^i_3+a^i_4}{2} \right] \\ \frac{(1+\theta _{r,i})(a^i_4-x)}{a^i_4-a^i_3+2\theta _{r,i}(a^i_4-x)}, &{}\text { if } x \in \left( \frac{a^i_3+a^i_4}{2},a^i_4 \right] . \end{array}\right. } \end{aligned}$$

Write $\varsigma =\sum \nolimits _{i=1}^n k_i \varsigma _i$. If $\alpha < 0.5$, then we have

${\tilde{Cr}} \{\varsigma \le u \} =\frac{1}{2}(1+ \sup \nolimits _{x \le u} \mu _{\varsigma }(x)- \sup \nolimits _{x> u} \mu _{\varsigma }(x))=\frac{1}{2}(1+ 1- \sup \nolimits _{x > u} \mu _{\varsigma }(x))$.

Therefore, ${\tilde{Cr}} \{\varsigma \le u \} \ge \alpha$ is equivalent to $\sup \nolimits _{x > u} \mu _{\varsigma }(x) \le 2-2\alpha$.

If we define $\varsigma _{sup}(\alpha )= sup\{r | \sup \nolimits _{x > u} \mu _{\varsigma }(x)\ge \alpha \}$ for $\alpha \in (0, 1]$, then we have $\varsigma _{sup}(2-2\alpha )\le u$.

Since, $\varsigma _1, \varsigma _2,..., \varsigma _n$ are mutually independent, we have

$\varsigma _{sup}(2-2\alpha ) ={\left( \sum \nolimits _{i=1}^n k_i \varsigma _i\right) }_{sup}(2-2\alpha )=\sum \nolimits _{i=1}^n k_i \varsigma _{i,sup}(2-2\alpha ) \le u$.

Note that $\mu _{\varsigma _i}(\frac{a^i_3+a^i_4}{2})=0.5$. If $2-2\alpha \ge 0.5$, i.e., $\alpha \in (0.5, 0.75]$, then for each i, $\varsigma _{i,sup}(2-2\alpha )$ is the solution of the following equation: $\frac{(-1+\theta _{l,i})x-\theta _{l,i} a^i_3+a^i_4}{a^i_4-a^i_3+2\theta _{l,i}(x-a^i_3)}=2-2\alpha$.

Solving the above equation, we have $\varsigma _{i,sup}(2-2\alpha )= \frac{(2\alpha -1)k_i a^i_4+(2(1-\alpha ) + (3-4\alpha )\theta _{l,i}) k_i a^i_3}{1+(3-4\alpha ) \theta _{l,i}}$.

Therefore, when $\alpha \in (0.5, 0.75]$, ${\tilde{Cr}} \{ \sum \nolimits _{i=1}^n k_i \varsigma _i \le u \} \ge \alpha$ is equivalent to

$$\begin{aligned} \sum ^n_{i=1} \frac{(2\alpha -1)k_i a^i_4+(2(1-\alpha ) + (3-4\alpha )\theta _{l,i}) k_i a^i_3}{1+(3-4\alpha ) \theta _{l,i}} \le u. \end{aligned}$$

On the other hand, if $2-2\alpha < 0.5$, i.e., $\alpha \in (0.75, 1]$, then for each i, $\varsigma _{i,sup}(2-2\alpha )$ is the solution of the following equation: $\frac{(1+\theta _{r,i})(a^i_4-x)}{a^i_4-a^i_3+2\theta _{r,i}(a^i_4-x)}=2-2\alpha .$

Solving the above equation gives $\varsigma _{i,sup}(2-2\alpha )= \frac{(2\alpha -1+(4\alpha -3)\theta _{r,i})k_i a^i_4+2(1-\alpha ) k_i a^i_3}{1+(4\alpha -3) \theta _{r,i}}$.

Therefore, when $\alpha \in (0.75, 1]$, ${\tilde{Cr}} \{ \sum \nolimits _{i=1}^n k_i \varsigma _i \le u \} \ge \alpha$ is equivalent to

$$\begin{aligned} \sum ^n_{i=1} \frac{(2\alpha -1+(4\alpha -3)\theta _{r,i})k_i a^i_4+2(1-\alpha ) k_i a^i_3}{1+(4\alpha -3) \theta _{r,i}} \le u. \end{aligned}$$

The proof of the theorem is complete. $\square$

Corollary 1

Let $\varsigma _i$ be the reduction of the type-2 trapezoidal fuzzy variable $\tilde{\varsigma _i} = (a^i_1, a^i_2, a^i_3, a^i_4; \theta _{l,i}, \theta _{r,i})$ obtained by the CV reduction method for $i=1,2,...,n$. Suppose $\varsigma _1,\varsigma _2,...,\varsigma _n$ are mutually independent, and $k_i \ge 0$ for $i=1,2,...,n$. The different ranges of $\alpha$ of generalized credibility, then ${\tilde{Cr}} \{ \sum \nolimits _{i=1}^n k_i \varsigma _i \ge u \} \ge \alpha$ is equivalent to

$$\begin{aligned}&\sum ^n_{i=1} \frac{(1-2\alpha +(1-4\alpha )\theta _{l,i})k_i a^i_4+2\alpha k_i a^i_3}{1+(1-4\alpha ) \theta _{l,i}}\\&\ge u \qquad \text { if }\alpha \in (0, 0.25]\\&\sum ^n_{i=1} \frac{(1-2\alpha )k_i a^i_4+(2\alpha + (4\alpha -1)\theta _{r,i}) k_i a^i_3}{1+(4\alpha -1) \theta _{r,i}}\\&\ge u \qquad \text { if }\alpha \in (0.25, 0.5]\\&\sum ^n_{i=1} \frac{(2\alpha -1)k_i a^i_1+(2(1-\alpha ) + (3-4\alpha )\theta _{r,i}) k_i a^i_2}{1+(3-4\alpha ) \theta _{r,i}}\\&\ge u \qquad \text { if }\alpha \in (0.5, 0.75]\\&\sum ^n_{i=1} \frac{(2\alpha -1+(4\alpha -3)\theta _{l,i})k_i a^i_1+2(1-\alpha ) k_i a^i_2}{1+(4\alpha -3) \theta _{l,i}}\\&\ge u \qquad \text { if }\alpha \in (0.75, 1]. \end{aligned}$$

Proof

Now, ${\tilde{Cr}} \{ \sum \nolimits _{i=1}^n k_i \varsigma _i \ge u \} \ge \alpha$ $\implies$ ${\tilde{Cr}} \{ \sum \nolimits _{i=1}^n -k_i \varsigma _i \le -u \} \ge \alpha$ $\implies$ ${\tilde{Cr}} \{ \sum \nolimits _{i=1}^n k_i \varsigma ^{\prime }_i \le u^{\prime } \} \ge \alpha$, where $\varsigma ^{\prime }_i=-\varsigma _i$ is the reduction of $-{\tilde{\varsigma }}_i=(-a^i_4, -a^i_3, -a^i_2, -a^i_1; \theta _{r,i}, \theta _{l,i})$ and $u^{\prime }=-u$. Then using Theorem 2, if $\alpha \in (0, 0.25]$, then the expression ${\tilde{Cr}} \{ \sum \nolimits _{i=1}^n k_i \varsigma ^{\prime }_i \le u^{\prime } \} \ge \alpha$ is equivalent to

$$\begin{aligned}&\sum ^n_{i=1} \frac{(1-2\alpha +(1-4\alpha )\theta _{l,i})k_i (-a^i_4)+2\alpha k_i (-a^i_3)}{1+(1-4\alpha ) \theta _{l,i}} \le -u\\&\quad \implies \sum ^n_{i=1} \frac{(1-2\alpha +(1-4\alpha )\theta _{l,i})k_i a^i_4+2\alpha k_i a^i_3}{1+(1-4\alpha ) \theta _{l,i}} \ge u. \end{aligned}$$

The equivalent expressions for other values of $\alpha$ are similarly obtained. $\square$

Theorem 3

(Qin et al. 2011) Let ${\tilde{\varsigma }}$ be a gamma T2FV define as ${\tilde{\gamma }}=(\lambda , r; \theta _l, \theta _r)$. Then we have,

(i) The reduction $\varsigma _1$ of ${\tilde{\varsigma }}$ has the following possibility distribution, when we use optimistic CV reduction method

$$\begin{aligned} \mu _{\varsigma _1}(x)= {\left\{ \begin{array}{ll} \frac{{ (1+\theta _r) \left( \frac{x}{\lambda r} \right) }^r exp \left( r-\frac{x}{\lambda } \right) }{1+\theta _r { \left( \frac{x}{\lambda r} \right) }^r exp \left( r-\frac{x}{\lambda } \right) }, &{} \text {if} \ {\left( \frac{x}{\lambda r}\right) }^r exp \left( r-\frac{x}{\lambda } \right) \le \frac{1}{2} \\ \frac{{\theta _r+(1-\theta _r) \left( \frac{x}{\lambda r} \right) }^r exp \left( r-\frac{x}{\lambda } \right) }{1+\theta _r-\theta _r { \left( \frac{x}{\lambda r} \right) }^r exp \left( r-\frac{x}{\lambda } \right) }, &{} \text {if} \ {\left( \frac{x}{\lambda r}\right) }^r exp \left( r-\frac{x}{\lambda } \right) > \frac{1}{2}. \end{array}\right. } \end{aligned}$$

(29)

(ii) The reduction $\varsigma _2$ of ${\tilde{\varsigma }}$ has the following possibility distribution, when we use pessimistic CV reduction method

$$\begin{aligned} \mu _{\varsigma _2}(x)= {\left\{ \begin{array}{ll} \frac{{ \left( \frac{x}{\lambda r} \right) }^r exp \left( r-\frac{x}{\lambda } \right) }{1+\theta _l { \left( \frac{x}{\lambda r} \right) }^r exp \left( r-\frac{x}{\lambda } \right) }, &{} \text {if} \ {\left( \frac{x}{\lambda r}\right) }^r exp \left( r-\frac{x}{\lambda } \right) \le \frac{1}{2} \\ \frac{{ \left( \frac{x}{\lambda r} \right) }^r exp \left( r-\frac{x}{\lambda } \right) }{1+\theta _l-\theta _l { \left( \frac{x}{\lambda r} \right) }^r exp \left( r-\frac{x}{\lambda } \right) }, &{} \text {if} \ {\left( \frac{x}{\lambda r}\right) }^r exp \left( r-\frac{x}{\lambda } \right) > \frac{1}{2}. \end{array}\right. } \end{aligned}$$

(30)

(iii) The reduction $\varsigma _3$ of ${\tilde{\varsigma }}$ has the following possibility distribution, when we use CV reduction method

$$\begin{aligned} \mu _{\varsigma _3}(x)= {\left\{ \begin{array}{ll} \frac{{(1+\theta _r) \left( \frac{x}{\lambda r} \right) }^r exp \left( r-\frac{x}{\lambda } \right) }{1+2\theta _r { \left( \frac{x}{\lambda r} \right) }^r exp \left( r-\frac{x}{\lambda } \right) }, &{} \text {if} \ {\left( \frac{x}{\lambda r}\right) }^r exp \left( r-\frac{x}{\lambda } \right) \le \frac{1}{2} \\ \frac{{\theta _l+(1-\theta _l) \left( \frac{x}{\lambda r} \right) }^r exp \left( r-\frac{x}{\lambda } \right) }{1+2\theta _l-2\theta _l { \left( \frac{x}{\lambda r} \right) }^r exp \left( r-\frac{x}{\lambda } \right) }, &{} \text {if} \ {\left( \frac{x}{\lambda r}\right) }^r exp \left( r-\frac{x}{\lambda } \right) > \frac{1}{2}. \end{array}\right. } \end{aligned}$$

(31)

Figures 8, 9 and 10 represent the possibility distribution of $\varsigma _1$, $\varsigma _2$ and $\varsigma _3$, respectively, defined in Theorem 3.

Fig. 8

The possibility distributions of $\mu _{\varsigma (x)}$, a type-2 gamma fuzzy number, using optimistic reduction method

Fig. 9

The possibility distributions of $\mu _{\varsigma (x)}$, a type-2 gamma fuzzy number, using pessimistic reduction method

Fig. 10

The possibility distributions of $\mu _{\varsigma (x)}$, a type-2 gamma fuzzy number, using CV reduction method

Defuzzification method for gamma T2FV

Due to the three-dimensional structure, computational complexity with T2FV is very high. A general idea to reduce its complexity is to convert a T2FV into a T1FV so that the methodologies to deal with T1FVs can also be applied to T2FVs. Qin et al. (2011) proposed a CV-bRT which reduces a T2FV to a type-1 fuzzy variable (T1FV) which may or may not be normal. In the following, we have presented a defuzzification method based on the CV-reduction for a gamma T2FV.

Theorem 4

(Sengupta et al. 2018) Let $\varsigma _i$ be the reduction of the Type-2 gamma fuzzy variable ${\tilde{\varsigma }}_i={\tilde{\gamma }}(\lambda _i, r_i ; \theta _{l,i}, \theta _{r,i})$, obtained by CV reduction method for $i = 1, 2, 3, ..., n$. Suppose $\varsigma _1, \varsigma _2, ..., \varsigma _n$ are mutually independent and $k_i \ge 0$ for $i = 1, 2, 3,..., n$. The different ranges of $\alpha$ of generalized credibility, then ${\tilde{Cr}} \{ \sum \nolimits _{i=1}^n k_i \varsigma _i \le u \} \ge \alpha$ is equivalent to

$$\begin{aligned}&\sum _{i=1}^n k_i(-\lambda _ir_i)W \left[ -\frac{1}{e}(2\alpha )^\frac{1}{r_i} \left[ 1+(1\right. \right. \\&\quad \left. \left. -4\alpha )\theta _{r,i}\right] ^{-\frac{1}{r_i}} \right] \le u, \text { if }\alpha \in (0, 0.25] \\&\sum _{i=1}^n k_i(-\lambda _ir_i)W \left[ -\frac{1}{e}[2\alpha +(4\alpha -1)\theta _{l,i}]^\frac{1}{r_i} \left[ 1\right. \right. \\&\quad \left. \left. +(4\alpha -1)\theta _{l,i}\right] ^{-\frac{1}{r_i}} \right] \le u, \text { if }\alpha \in (0.25, 0.5]\\&\sum _{i=1}^n k_i(-\lambda _ir_i)W \left[ -\frac{1}{e}[2(1-\alpha )+(3-4\alpha )\theta _{l,i}]^\frac{1}{r_i} \left[ 1+(3\right. \right. \\&\quad \left. \left. -4\alpha )\theta _{l,i}\right] ^{-\frac{1}{r_i}} \right] \le u, \text { if }\alpha \in (0.5, 0.75]\\&\sum _{i=1}^n k_i(-\lambda _ir_i)W \left[ -\frac{1}{e}[2(1-\alpha )]^\frac{1}{r_i} \left[ 1+(4\alpha \right. \right. \\&\quad \left. \left. -3)\theta _{r,i}\right] ^{-\frac{1}{r_i}} \right] \le u, \text { if }\alpha \in (0.75, 1], \end{aligned}$$

where W is called as product log function and defined as follows: $W(z)=\sum \nolimits ^{\infty }_{n=1}\frac{(-1)^{n-1}}{(n-1)!}z^n$.

Corollary 2

Let $\varsigma _i$ be the reduction of the Type-2 gamma fuzzy variable ${\tilde{\varsigma }}_i={\tilde{\gamma }}(\lambda _i, r_i ; \theta _{l,i}, \theta _{r,i})$, obtained by CV reduction method for $i = 1, 2, 3, ..., n$. Suppose $\varsigma _1, \varsigma _2, ..., \varsigma _n$ are mutually independent and $k_i \ge 0$ for $i = 1, 2, 3, ..., n$. From Theorem 4, the different ranges of $\alpha$ of generalized credibility, then ${\tilde{Cr}} \{ \sum \nolimits _{i=1}^n k_i \varsigma _i \ge u \} \ge \alpha$ is equivalent to

$$\begin{aligned}&\sum _{i=1}^n k_i(-\lambda _ir_i)W \left[ -\frac{1}{e}(2\alpha )^{-\frac{1}{r_i}} \left[ 1+(1\nonumber \right. \right. \\&\quad \left. \left. -4\alpha )\theta _{l,i}\right] ^{\frac{1}{r_i}} \right] \ge u. \text { if }\alpha \in (0, 0.25] \\&\sum _{i=1}^n k_i(-\lambda _ir_i)W \left[ -\frac{1}{e}[2\alpha +(4\alpha -1)\theta _{r,i}]^{-\frac{1}{r_i}} \left[ 1+(4\alpha \nonumber \right. \right. \\&\quad \left. \left. -1)\theta _{r,i}\right] ^{\frac{1}{r_i}} \right] \ge u \text { if }\alpha \in (0.25, 0.5]\\&\sum _{i=1}^n k_i(-\lambda _ir_i)W \left[ -\frac{1}{e}[2(1-\alpha )+(3-4\alpha )\theta _{r,i}]^{-\frac{1}{r_i}} \left[ 1+(3\nonumber \right. \right. \\&\quad \left. \left. -4\alpha )\theta _{r,i}\right] ^{\frac{1}{r_i}} \right] \ge u\text { if }\alpha \in (0.5, 0.75]\\&\sum _{i=1}^n k_i(-\lambda _ir_i)W \left[ -\frac{1}{e}[2(1-\alpha )]^{-\frac{1}{r_i}} \left[ 1+(4\alpha \nonumber \right. \right. \\&\quad \left. \left. -3)\theta _{l,i}\right] ^{\frac{1}{r_i}} \right] \ge u \text { if }\alpha \in (0.75, 1]. \end{aligned}$$

Proof

If we considered the mean $\mu _i=\lambda _ir_i$ and the standard deviation $\sigma ^2_i=\lambda _ir_i^2$, then all the expressions for $\alpha \in (0,0.25]$ of Theorem 4 can be written as,

$$\begin{aligned}&\sum _{i=1}^n -k_i\mu _iW \left[ -\frac{1}{e}{(2\alpha )^\frac{\mu _i}{\sigma ^2_i} \left[ 1+(1-4\alpha )\theta _{r,i}\right] ^{-\frac{\mu _i}{\sigma ^2_i}}} \right] \le u. \end{aligned}$$

Now, ${\tilde{Cr}} \{ \sum \nolimits _{i=1}^n k_i \varsigma _i \ge u \} \ge \alpha$ $\implies$ ${\tilde{Cr}} \{ \sum \nolimits _{i=1}^n -k_i \varsigma _i \le -u \} \ge \alpha$ $\implies$ ${\tilde{Cr}} \{ \sum \limits _{i=1}^n k_i \varsigma ^{\prime }_i \le u^{\prime } \} \ge \alpha$, where $\varsigma ^{\prime }_i=-\varsigma _i$ is the reduction of $-{\tilde{\varsigma }}_i={\tilde{\gamma }}(\frac{\mu ^2_i}{\sigma ^2_i}, -\frac{\sigma ^2_i}{\mu _i} ; \theta _{r,i}, \theta _{l,i})$ and $u^{\prime }=-u$. Here, for $\varsigma ^{\prime }_i$, the mean $\mu _i$ is taken negative and left and right spreads are interchanged. This can also be explained geometrically. Then, using above theorem, if $\alpha \in (0, 0.25]$, then the expression ${\tilde{Cr}} \{ \sum \nolimits _{i=1}^n k_i \varsigma ^{\prime }_i \le u^{\prime } \} \ge \alpha$ is equivalent to

$$\begin{aligned}&\sum _{i=1}^n k_i\mu _iW \left[ -\frac{1}{e}{(2\alpha )^{-\frac{\mu _i}{\sigma ^2_i}} \left[ 1+(1-4\alpha )\theta _{l,i}\right] ^{\frac{\mu _i}{\sigma ^2_i}}} \right] \le -u\\ \implies&\sum _{i=1}^n -k_i\mu _iW \left[ -\frac{1}{e}{(2\alpha )^{-\frac{\mu _i}{\sigma ^2_i}} \left[ 1+(1-4\alpha )\theta _{l,i}\right] ^{\frac{\mu _i}{\sigma ^2_i}}} \right] \ge u. \end{aligned}$$

Now putting the values $\mu _i=\lambda _ir_i$ and $\sigma ^2_i=\lambda _ir_i^2$, we can rewrite the above expression as

$$\begin{aligned} \sum _{i=1}^n k_i(-\lambda _ir_i)W \left[ -\frac{1}{e}{(2\alpha )^{-\frac{1}{r_i}} \left[ 1+(1-4\alpha )\theta _{l,i}\right] ^{\frac{1}{r_i}}} \right] \ge u. \end{aligned}$$

The expressions for other values of $\alpha$ are similarly obtained. $\square$