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

A Correction to this article was published on 29 November 2021

This article has been updated

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.

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Availability of data and material

All data are real-life, which we collect from local and Internet.

Change history

Notes

  1. 1.

    https://google.co.in/maps.

  2. 2.

    http://www.balajifreightlogistics.com/vehicle-list-dimension.html.

References

  1. Abualigah L, Diabat A (2021) Advances in sine cosine algorithm: a comprehensive survey. Artif Intell Rev. pp 1–42

  2. Abualigah L, Diabat A, Mirjalili S, Abd Elaziz M, Gandomi AH (2021) The arithmetic optimization algorithm. Comput Methods Appl Mech Eng 376:113609

    MathSciNet  Article  Google Scholar 

  3. Abualigah LMQ et al (2019) Feature selection and enhanced krill herd algorithm for text document clustering. Springer, Berlin

    Book  Google Scholar 

  4. Abualigah L, Yousri D, Abd Elaziz M, Ewees AA, Al-qaness MA, Gandomi AH (2021) Aquila optimizer: a novel meta-heuristic optimization algorithm. Comput Ind Eng 157:107250

    Article  Google Scholar 

  5. Arqub OA (2017) Adaptation of reproducing kernel algorithm for solving fuzzy fredholm-volterra integrodifferential equations. Neural Comput Appl 28(7):1591–1610

    Article  Google Scholar 

  6. Arqub OA, Abo-Hammour Z (2014) Numerical solution of systems of second-order boundary value problems using continuous genetic algorithm. Inf Sci 279:396–415

    MathSciNet  Article  Google Scholar 

  7. Arqub OA, Al-Smadi M (2020) Fuzzy conformable fractional differential equations: novel extended approach and new numerical solutions. Soft Comput. pp 1–22

  8. Arqub OA, Al-Smadi M, Momani S, Hayat T (2017) Application of reproducing kernel algorithm for solving second-order, two-point fuzzy boundary value problems. Soft Comput 21(23):7191–7206

    Article  Google Scholar 

  9. Arqub OA, Mohammed A-S, Momani S, Hayat T (2016) Numerical solutions of fuzzy differential equations using reproducing kernel hilbert space method. Soft Comput 20(8):3283–3302

    Article  Google Scholar 

  10. Baidya A, Bera UK, Maiti M (2015) Breakable fuzzy multi-stage transportation problem. J Op Res Soc China 3(1):53–67

    MathSciNet  Article  Google Scholar 

  11. Bera S, Giri PK, Jana DK, Basu K, Maiti M (2020) Fixed charge 4d-tp for a breakable item under hybrid random type-2 uncertain environments. Inf Sci

  12. Calvete HI, Galé C, Iranzo JA (2016) An improved evolutionary algorithm for the two-stage transportation problem with fixed charge at depots. OR Spectr 38(1):189–206

    MathSciNet  Article  Google Scholar 

  13. Cosma O, Dănciulescu D, Pop PC (2019) On the two-stage transportation problem with fixed charge for opening the distribution centers. IEEE Access 7:113684–113698

    Article  Google Scholar 

  14. Das A, Bera UK, Maiti M (2018) Defuzzification and application of trapezoidal type-2 fuzzy variables to green solid transportation problem. Soft Comput 22(7):2275–2297

    Article  Google Scholar 

  15. Dutta A, Jana DK (2017) Expectations of the reductions for type-2 trapezoidal fuzzy variables and its application to a multi-objective solid transportation problem via goal programming technique. J Uncertain Anal Appl 5(1):3

    Article  Google Scholar 

  16. Giri PK, Maiti MK, Maiti M (2014) Fuzzy stochastic solid transportation problem using fuzzy goal programming approach. Comput Ind Eng 72:160–168

    Article  Google Scholar 

  17. Goldberg DE, Samtani MP (1986) Engineering optimization via genetic algorithm. Electr Comput, ASCE 471–482

  18. Halder S, Das B, Panigrahi G, Maiti M (2017) Some special fixed charge solid transportation problems of substitute and breakable items in crisp and fuzzy environments. Comput Ind Eng 111:272–281

    Article  Google Scholar 

  19. Haley K (1962) New methods in mathematical programming–the solid transportation problem. Oper Res 10(4):448–463

    Article  Google Scholar 

  20. Hirsch WM, Dantzig GB (1968) The fixed charge problem. Naval Res Logist Q 15(3):413–424

    MathSciNet  Article  Google Scholar 

  21. Hitchcock FL (1941) The distribution of a product from several sources to numerous localities. J Math Phys 20(1–4):224–230

    MathSciNet  Article  Google Scholar 

  22. Koopmans TC (1949) Optimum utilization of the transportation system. Econom J Econom Soc. pp 136–146

  23. Kundu P, Kar S, Maiti M (2014) Fixed charge transportation problem with type-2 fuzzy variables. Inf Sci 255:170–186

    MathSciNet  Article  Google Scholar 

  24. Kundu P, Kar S, Maiti M (2015) Multi-item solid transportation problem with type-2 fuzzy parameters. Appl Soft Comput 31:61–80

    Article  Google Scholar 

  25. Kwakernaak H (1978) Fuzzy random variables–i. definitions and theorems. Inf Sci 15(1):1–29

    MathSciNet  Article  Google Scholar 

  26. Liu F (2008) An efficient centroid type-reduction strategy for general type-2 fuzzy logic system. Inf Sci 178(9):2224–2236

    MathSciNet  Article  Google Scholar 

  27. Liu Z-Q, Liu Y-K (2007) Fuzzy possibility space and type-2 fuzzy variable. 2007 IEEE Symposium on Foundations of Computational Intelligence, IEEE, pp. 616–621

  28. Liu Z-Q, Liu Y-K (2010) Type-2 fuzzy variables and their arithmetic. Soft Comput 14(7):729–747

    Article  Google Scholar 

  29. Mollanoori H, Tavakkoli-Moghaddam R, Triki C, Hajiaghaei-Keshteli M, Sabouhi F (2019) Extending the solid step fixed-charge transportation problem to consider two-stage networks and multi-item shipments. Comput Ind Eng 137:106008

    Article  Google Scholar 

  30. Pramanik S, Jana DK, Mondal SK, Maiti M (2015) A fixed-charge transportation problem in two-stage supply chain network in gaussian type-2 fuzzy environments. Inf Sci 325:190–214

    MathSciNet  Article  Google Scholar 

  31. Qin R, Liu Y-K, Liu Z-Q (2011) Methods of critical value reduction for type-2 fuzzy variables and their applications. J Comput Appl Math 235(5):1454–1481

    MathSciNet  Article  Google Scholar 

  32. Sengupta D, Bera UK, Datta A, Das A (2017) Reduction method of zigzag type-2 uncertain variable and its application in two stage stp. 2017 5th International Symposium on Computational and Business Intelligence (ISCBI), IEEE, pp. 139–143

  33. Sengupta D, Das A, Bera UK (2018) A gamma type-2 defuzzification method for solving a solid transportation problem considering carbon emission. Appl Intell 48(11):3995–4022

    Article  Google Scholar 

  34. Shell E (1955) Distribution of a product by several properties, directorate of management analysis. Proceedings of the second symposium in linear programming 2:615–642

  35. Singh A, Bera UK, Sengupta D, Datta A, Das A (2017) Defuzzication method of type-2 gamma fuzzy variables and its application to transportation problem. 2017 5th International Symposium on Computational and Business Intelligence (ISCBI), IEEE, pp. 144–149

  36. Wang C, Li Y, Huang G (2017) Taguchi-factorial type-2 fuzzy random optimization model for planning conjunctive water management with compound uncertainties. Environ Model Softw 97:184–200

    Article  Google Scholar 

  37. Yang L, Liu P, Li S, Gao Y, Ralescu DA (2015) Reduction methods of type-2 uncertain variables and their applications to solid transportation problem. Inf Sci 291:204–237

    MathSciNet  Article  Google Scholar 

  38. Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353

    Article  Google Scholar 

  39. Zadeh LA (1968) Probability measures of fuzzy events. J Math Anal Appl 23(2):421–427

    MathSciNet  Article  Google Scholar 

  40. Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning–i. Inf Sci 8(3):199–249

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The authors would like to express their gratitude to the anonymous referees for carefully reading the paper and their helpful comments and suggestions, which greatly improved the quality of the paper.

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Correspondence to Sudeshna Devnath.

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Appendices

Background and preliminaries

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 \(\sum \nolimits _{i=1}^{\infty }p_i=1\).

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:

  1. (i)

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

  2. (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 \(\mathbb {R}\).

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 \(0 < \lambda \le \frac{1}{r}\).

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.

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 \({\tilde{\gamma }} ({\hat{\lambda }}, {\hat{r}}; \hat{\theta _{l}}, \hat{\theta _{r}})\).

Critical values (CV) for regular fuzzy variable

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

  1. (i)

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

  2. (ii)

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

  3. (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
figure5

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

Fig. 6
figure6

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

Fig. 7
figure7

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
figure8

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

Fig. 9
figure9

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

Fig. 10
figure10

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. \(\square \)

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Devnath, S., Giri, P.K., Mondal, S.S. et al. Multi-item two-stage fixed-charge 4DTP with hybrid random type-2 fuzzy variable. Soft Comput 25, 15083–15114 (2021). https://doi.org/10.1007/s00500-021-06371-3

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Keywords

  • Two-stage multi-item 4D transportation problems
  • Random type-2 trapezoidal and gamma fuzzy variables
  • CV-based reduction method
  • Generalized credibility measure technique
  • CV-centroid method