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Bi-Level linear programming of intuitionistic fuzzy

  • Fuzzy systems and their mathematics
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Abstract

In this research paper, we will solve problems of Bi-level linear fractional programming (BL-LFP) by proposing an interactive approach. Based on the imposition of the relationship DM. to obtained adequate solution, DMs will updating the minimal adequate level at upper level permanently, and that is through. Firstly, the decision makers uncertainty is described by introducing the membership function and non-membership function. Secondly, the opting of minimum adequate degree, leads to obtain the adequate solution, with the overall adequate balance considerations among two levels. For more, this paper gives an algorithm of the proposed approach. At the end, we give a numerical example to explain the feasibility of that approach.

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Acknowledgements

This research was funded by the Deanship of Scientific Research at Princess Nourah bint Abdulrahman University through the Fast-track Research Funding Program.

Funding

This research was funded by the Deanship of Scientific Research at Princess Nourah bint Abdulrahman University through the Fast−track Research Funding Program.

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Authors and Affiliations

  1. Department of Mathematical Science, Faculty of Science, Princess Nourah Bint Abdulrahman University, Riyadh, Saudi Arabia

    Nazek A. Alessa

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  1. Nazek A. Alessa
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Correspondence to Nazek A. Alessa.

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Appendix

Here we will consider an example clarifies Tri-level linear fractional programming problem (TL-LFPP) as:

$$ \mathop {\min }\limits_{{w_{1} ,w_{2} ,w _{3} }} X_{1} \left( w \right) = { }\frac{{7w_{1} + 6w_{2} + 8w_{3} }}{{9w_{1} + 10w_{2} + 8w_{3} + 2}} $$
$$ \mathop {\min }\limits_{{w_{2} ,w_{3} }} X_{2} \left( w \right) = { }\frac{{8w_{1} + 7w_{2} + 9w_{3} }}{{8w_{1} + 9w_{2} + 6w_{3} + 1.5}} $$
$$ \mathop {\min }\limits_{{w_{3} }} X_{3} \left( w \right) = { }\frac{{9w_{1} + 8w_{2} + 10w_{3} }}{{7w_{1} + 8w_{2} + 4w_{3} + 1}} $$
$$ \begin{gathered} {\text{s.t}}{.}\;\;w \, = \left( {w_{1} , \, w_{2} , \, w_{3} } \right)\in V = \, \{ \left( {w_{1} , \, w_{2} , \, w_{3} } \right) \, | \, 3w_{1} + \, 2w_{2} + \, 4w_{3} < \, 25, \hfill \\ \;\;\;\;\;\;\;4w_{1} + 3w_{2} + 5w_{3} \le \, 28 \, , \, 5w_{1} + \, 4w_{2} + \, 6w_{3} \le \, 30, \hfill \\ \;\;\;\;\;\;\;2w_{1} + \, 1.5w_{2} + \, 3w_{3} \le \, 24 \, , \, 6w_{1} + \, 4.5w_{2} + \, 7w_{3} \le \, 32, \hfill \\ \;\;\;\;\;\;\;4w_{1} + \, 2w_{2} + \, 2w_{3} \le \, 18 \, , \, 5w_{1} + \, 3w_{2} + \, 3w_{3} \le \, 20, \hfill \\ \;\;\;\;\;\;\;6w_{1} + \, 4w_{2} + \, 4w_{3} \le \, 22 \, , \, 3w_{1} + \, 1.5w_{2} + \, 1.5w_{3} \le \, 16, \hfill \\ \;\;\;\;\;\;\;7w_{1} + \, 4.5w_{2} + \, 4.5x_{3} \le \, 24 \, ,w_{1} + w_{2} + w_{3} \ge \, 0\} \hfill \\ \end{gathered} $$
(17)

By using a transformation method of Charnos and Cooper (Charnes and Cooper 1962), which was illustrated in detail in (Subject and Shiv 2016), to illustrate that TL-LFPP is equivalent to the following TL-LPP.

$$ \mathop {\min }\limits_{{c_{1} ,c_{2} ,c_{3} }} X_{1} \left( {c , \theta } \right) = 7c_{1} + 6c_{2} + 8c_{3} , $$
$$ \mathop {\min }\limits_{{{\text{c}}_{2} ,{\text{c}}_{3} }} X_{2} \left( {c , \theta } \right) = 8y_{1} + 7y_{2} + 9y_{3} , $$
$$ \mathop {\min }\limits_{{c_{3} }} {\text{X}}_{3} \left( {c , \theta } \right) = 9c_{1} + 8c_{2} + 10c_{3} , $$
$$ \begin{gathered} {\mathbf{s}}.{\mathbf{t}}. \left( {c \, , \, \tau } \right) \, = \, (c_{1} , \, c_{2} , \, c_{3} ,\theta ) \in G = \, \{ (c_{1} , \, c_{2} , \, c_{3} ,\theta )| \hfill \\ \;\;\;\;\;\;9c_{1} + \, 10c_{2} + \, 8c_{3} + \, 2 \theta \le 1 \, , \, 8c_{1} + \, 9c_{2} + \, 6c_{3} + \, 1.5\theta \le \, 1, \hfill \\ \;\;\;\;\;\;7c_{1} + \, 8c_{2} + \, 4c_{3} + \theta \le 1 \, , \, 3c_{1} + \, 2c_{2} + \, 4c_{3} {-}25\theta \le \, 0, \hfill \\ \;\;\;\;\;\;4c_{1} + \, 3c_{2} + \, 5c_{3} {-} \, 28\theta \le 0 \, , \, 5c_{1} + \, 4c_{2} + \, 6c_{3} {-} \, 30 \theta \le \, 0, \hfill \\ \;\;\;\;\;\;2c_{1} + \, 1.5c_{2} + \, 3c_{3} {-} \, 24\theta \le 0 \, , \, 6c_{1} + \, 4.5c_{2} + \, 7c_{3} {-} \, 32\theta \le \, 0, \hfill \\ \;\;\;\;\;\;4c_{1} + \, 2c_{2} + \, 2c_{3} {-} \, 18\theta \le 0 \, , \, 5c_{1} + \, 3c_{2} + \, 3c_{3} {-} \, 20 \theta \le \, 0, \hfill \\ \;\;\;\;\;\;6c_{1} + \, 4c_{2} + \, 4c_{3} {-} \, 22 \theta \le 0 \, , \, 3c_{1} + \, 1.5c_{2} + \, 1.5c_{3} {-}16\theta \le \, 0, \hfill \\ \;\;\;\;\;\;7c_{1} + \, 4.5c_{2} + \, 4.5c_{3} {-} \, 24\theta \le 0 \, ,c_{1} ,c_{2} ,c_{3} \ge \, 0 \, \theta , > 0\} \hfill \\ \end{gathered} $$
(18)

We choose ν1 = 10, ν2 = 6, ν3 = 2, γ1 = 0.95, γ2 = 0.85, [σ1L, σ1U] = [σ2L, σ2U] = [0.75, 0.9]. The solution of problem (15) is.

c = (8.581, 6.1466, 9.96877), θ = 1.6, ψ = 0.58112.

X1 = −516.22, X2 = –451.7, X3 = –371.435,

a1(X1) = 0.804 < γ1 = 0.95. a2(X2) = 0.595 < γ2 = 0.85.

a3(X3) = 0.581 < σ1 = 0.7412. σ2 = 0.9755.

Both of DM1 and DM2 aren’t satisfied with the above solution, so if DM2 changes γ2 = 0.85 to γ'2 = 0.65. So the corresponding problem of (16) is formulated as:

$$ \left\{ {\begin{array}{*{20}c} {\mathop {\max }\limits_{{\left( {c,\theta } \right),\psi }} \psi } \\ {{\mathbf{s}}.{\mathbf{t}} (c,\theta ) \, = \, (c_{1} ,c_{2} ,c_{3} ,\theta ) \in H,} \\ {(X_{1} (c \, ,\theta )) \, \ge \psi ,} \\ {a_{2} (X_{2} (c \, ,\theta )) \, \ge \, 0.65 \, ,} \\ {a_{3} (X_{3} (c{ ,}\theta )) \, \ge \, \psi \, ,} \\ {\psi \in \left[ {0,1} \right],} \\ {\lambda_{n} (X_{n} (c{ ,,}\theta )) \, \ge \, 0 , n = \, 1, \, 2, \, 3} \\ {\xi_{n} (X_{n} (c \, ,\theta )) \ge \lambda_{n} (X_{n} (c{ ,}\theta )) , n = \, 1, \, 2, \, 3} \\ {\xi_{n} (X_{n} (c \, ,\theta )) \ge \lambda_{n} (X_{n} (c{ ,}\theta )) \le 1, n = \, 1, \, 2, \, 33} \\ \end{array} } \right. $$
(19)

The solution of problem (19) is c = (8.791, 7.257, 8.916), θ = 2.1, ψ = 0.558775,

X1= −519.09, X2= −453.12, X3 = −371.4, a1(X1) = 0.865072 < γ1= 0.95,

a2(X2) = 0.65, a3(X3) = 0.558775, σ1 = 0.7514, σ2 = 0.8596.

The termination condition (7) isn't satisfied the value of a1(X1), if DM1 changes γ2 = 0.95 to γ'1 = 0.91 then solves the problem (20);

$$ \left\{ {\begin{array}{*{20}c} {\mathop {\max }\limits_{{\left( {c,\theta } \right),\psi }} \psi } \\ {{\mathbf{s}}.{\mathbf{t}} (c \, ,\theta ) \, = \, (c_{1} ,c_{2} ,c_{3} ,\theta ) \in H,} \\ {a_{1} (X_{1} (c \, ,\theta )) \, \ge \, 0.91 \, ,} \\ {a_{2} (X_{2} (c \, ,\theta )) \, \ge \, 0.65 \, ,} \\ {a_{3} (X_{3} (c \, ,\theta )) \, \ge y ,} \\ {\psi \in \left[ {0, \, 1} \right] \, ,} \\ {l\lambda_{n} (X_{n} (c \, ,\theta )) \, \ge \, 0 , n = \, 1, \, 2, \, 3} \\ {\xi_{n} (X_{n} (c \, ,\theta )) \ge \lambda_{n} (X_{n} (c \, ,\theta )) , n = \, 1, \, 2, \, 3} \\ {\xi_{n} (X_{n} (c \, ,\theta )) + \lambda_{n} (X_{n} (c \, ,\theta )) \le 1 \, , n = \, 1, \, 2, \, 3} \\ \end{array} } \right. $$
(20)

The solution of problem (14) is.

c = (8.873, 9.246, 7.917), θ = 1.513, ψ = 0.537895.

X1 = −518.12, X2 = −450.36, X3 = −369.57,

a1(X1) = 0.91, a2(X2) = 0.701, a3(X3) = 0.5379,

σ1 = 0.778453, σ2 = 0.767755.

By now, a1(X1) = 0.91 = γ'1, a2(X2) = 0.701 > 0.65 = γ'2,

Moreover σ1 = 0.778453, σ2 = 0.767755 are all in [0.75, 0.9]. This is meaning that, all the proposed algorithm's termination conditions are satisfied, the DMs find the adequate solution.

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Alessa, N.A. Bi-Level linear programming of intuitionistic fuzzy. Soft Comput 25, 8635–8641 (2021). https://doi.org/10.1007/s00500-021-05791-5

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