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Solving Systems of Linear Equations under Conditions of Uncertainty on the Example of the Leontief Model

Dariusz Kacprzak
Dariusz Kacprzak
   | 11. Sept. 2019
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Online veröffentlicht: 11. Sept. 2019
Seitenbereich: 244 - 259
DOI: https://doi.org/10.1515/ceej-2018-0017
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© 2018 D. Kacprzak, published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Fig. 1

Membership function of the triangular fuzzy number A = (a, b, c) and α-cutSource: Author's own elaboration.
Membership function of the triangular fuzzy number A = (a, b, c) and α-cutSource: Author's own elaboration.

Fig. 2

a) Ordered fuzzy number, b) OFN presented in relation to CFN, c) Arrow showing the order of the reverse functions and the orientation ofSource: (Kosiński and Prokopowicz, 2004).
a) Ordered fuzzy number, b) OFN presented in relation to CFN, c) Arrow showing the order of the reverse functions and the orientation ofSource: (Kosiński and Prokopowicz, 2004).

Fig. 3

a) OFN with positive orientation, b) OFN with negative orientationSource: (Kacprzak, 2010).
a) OFN with positive orientation, b) OFN with negative orientationSource: (Kacprzak, 2010).

Fig. 4

An example of an OFN together with its characteristic pointsSource: (Kacprzak, 2010).
An example of an OFN together with its characteristic pointsSource: (Kacprzak, 2010).

Fig. 5

Results obtained in Examples 3 and 4: a) constructed on a CFN, b) based on positively oriented OFNs, c) based on negatively oriented OFNsSource: Author's own elaboration
Results obtained in Examples 3 and 4: a) constructed on a CFN, b) based on positively oriented OFNs, c) based on negatively oriented OFNsSource: Author's own elaboration

Input-output table

Sectorinputs-outputsFinal outputTotal output
j
12…n
1x11x12…x1nd1X1
2x21x22…x2nd2X2
i⁝⁝⁝⋱⁝⁝⁝
nxn1xn2…xnndnXn

Total output level as dependent on the direction of the changes of the final output level

Lp.d˜$\widetilde{d}$X˜$\widetilde{X}$ΔdΔX
1.((23815,23815,23815)(22615,22615,22615))$\left( \begin{array}{*{35}{l}}\left( 23815,23815,23815 \right) \\\left( 22615,22615,22615 \right) \\\end{array} \right)$((28576.2,28576.2,28576.2)(24197.6,24197.6,24197.6))$\left( \begin{array}{*{35}{l}}\left( 28576.2,28576.2,28576.2 \right) \\\left( 24197.6,24197.6,24197.6 \right) \\\end{array} \right)$(00)$\left( \begin{array}{*{35}{l}}0 \\0 \\\end{array} \right)$(00)$\left( \begin{array}{*{35}{l}}0 \\0 \\\end{array} \right)$
2.((23815,23815,23815)(22125,22615,23105))$\left( \begin{array}{*{35}{l}}\left( 23815,23815,23815 \right) \\\left( 22125,22615,23105 \right) \\\end{array} \right)$((28557,28576.2,28595.4)(23686.7,24197.6,24708.5))$\left( \begin{matrix}\left( 28557,28576.2,28595.4 \right) \\\left( 23686.7,24197.6,24708.5 \right) \\\end{matrix} \right)$(0980)$\left( \begin{matrix}0 \\980 \\\end{matrix} \right)$(38.41021.7)$\left( \begin{matrix}38.4 \\1021.7 \\\end{matrix} \right)$
3.((23815,23815,23815)(23105,22615,22125))$\left( \begin{matrix}\left( 23815,23815,23815 \right) \\\left( 23105,22615,22125 \right) \\\end{matrix} \right)$((28557,28576.2,28595.4)23686.7,24197.6,24708.5)$\left( \begin{matrix}\left( 28557,28576.2,28595.4 \right) \\23686.7,24197.6,24708.5 \\\end{matrix} \right)$(0−980)$\left( \begin{matrix}0 \\-980 \\\end{matrix} \right)$(−38.4−1021.7)$\left( \begin{matrix}-38.4 \\-1021.7 \\\end{matrix} \right)$
4.((23305,23815,24325)(22615,22615,22615))$\left( \begin{array}{*{35}{l}}\left( 23305,23815,24325 \right) \\\left( 22615,22615,22615 \right) \\\end{array} \right)$((27983.2,28576.2,29169.2)(24184.3,24197.6,24210.9))$\left( \begin{matrix}\left( 27983.2,28576.2,29169.2 \right) \\\left( 24184.3,24197.6,24210.9 \right) \\\end{matrix} \right)$(10200)$\left( \begin{matrix}1020 \\0 \\\end{matrix} \right)$(118626.5)$\left( \begin{matrix}1186 \\26.5 \\\end{matrix} \right)$
5.((24325,23815,23305)(22615,22615,22615))$\left( \begin{matrix}\left( 24325,23815,23305 \right) \\\left( 22615,22615,22615 \right) \\\end{matrix} \right)$((27983.2,28576.2,29169.2)(24184.3,24197.6,24210.9))$\left( \begin{matrix}\left( 27983.2,28576.2,29169.2 \right) \\\left( 24184.3,24197.6,24210.9 \right) \\\end{matrix} \right)$(−10200)$\left( \begin{matrix}-1020 \\0 \\\end{matrix} \right)$(−1186−26.5)$\left( \begin{matrix}-1186 \\-26.5 \\\end{matrix} \right)$
6.((23305,23815,24325)(22125,22615,23105))$\left( \begin{matrix}\left( 23305,23815,24325 \right) \\\left( 22125,22615,23105 \right) \\\end{matrix} \right)$((27964,28576.2,29188.4)(23673.5,23197.6,24721.7))$\left( \begin{matrix}\left( 27964,28576.2,29188.4 \right) \\\left( 23673.5,23197.6,24721.7 \right) \\\end{matrix} \right)$(1020980)$\left( \begin{matrix}1020 \\980 \\\end{matrix} \right)$(1224.41048.3)$\left( \begin{matrix}1224.4 \\1048.3 \\\end{matrix} \right)$
7.((23305,23815,24325)(23105,22615,22125))$\left( \begin{array}{*{35}{l}}\left( 23305,23815,24325 \right) \\\left( 23105,22615,22125 \right) \\\end{array} \right)$((28002.4,28576.2,29150)(24695.2,24197.6,23700))$\left( \begin{array}{*{35}{l}}\left( 28002.4,28576.2,29150 \right) \\\left( 24695.2,24197.6,23700 \right) \\\end{array} \right)$(1020−980)$\left( \begin{array}{*{35}{l}}1020 \\-980 \\\end{array} \right)$(1147.5−995.2)$\left( \begin{array}{*{35}{l}}1147.5 \\-995.2 \\\end{array} \right)$
8.((24325,23815,23305)(22125,22615,23105))$\left( \begin{array}{*{35}{l}}\left( 24325,23815,23305 \right) \\\left( 22125,22615,23105 \right) \\\end{array} \right)$((29150,28576.2,28002.4)(23700,24197.6,24695.2))$\left( \begin{array}{*{35}{l}}\left( 29150,28576.2,28002.4 \right) \\\left( 23700,24197.6,24695.2 \right) \\\end{array} \right)$(−1020    980)$\left( \begin{array}{*{35}{l}}-1020 \\\,\,\,\,980 \\\end{array} \right)$(−1147.5   995.2)$\left( \begin{array}{*{35}{l}}-1147.5 \\\,\,\,995.2 \\\end{array} \right)$
9.((24325,23815,23305)(23105,22615,22125))$\left( \begin{array}{*{35}{l}}\left( 24325,23815,23305 \right) \\\left( 23105,22615,22125 \right) \\\end{array} \right)$((29188.4,28576.2,27964)(24721.7,24197.6,23673.5))$\left( \begin{array}{*{35}{l}}\left( 29188.4,28576.2,27964 \right) \\\left( 24721.7,24197.6,23673.5 \right) \\\end{array} \right)$(−1020  −980)$\left( \begin{array}{*{35}{l}}-1020 \\\,\,-980 \\\end{array} \right)$(−1224.4−1048.3)$\left( \begin{array}{*{35}{l}}-1224.4 \\-1048.3 \\\end{array} \right)$
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