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

Dariusz Kacprzak

doi:10.1515/ceej-2018-0017

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

Central European Economic Journal

Volume 5 (2018): Issue 52 (January 2018)

By: Dariusz Kacprzak

Open Access

|Sep 2019

Figures & Tables

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

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

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

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

Tab. 1

Input-output table

Sector		inputs-outputs				Final output	Total output
				j
		1	2	…	n
	1	x₁₁	x₁₂	…	x_1n	d₁	X₁
	2	x₂₁	x₂₂	…	x_2n	d₂	X₂
i	⁝	⁝	⁝	⋱	⁝	⁝	⁝
	n	x_n1	x_n2	…	x_nn	d_n	X_n

[i] Source: Author's own elaboration based on (Gruszczyński and Podgórska, 2004, 371).

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

Tab. 2

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

Lp.	$\tilde{d}$	$\tilde{X}$	Δd	ΔX
1.	$(\begin{matrix} (23815, 23815, 23815) \\ (22615, 22615, 22615) \end{matrix})$	$(\begin{matrix} (28576.2, 28576.2, 28576.2) \\ (24197.6, 24197.6, 24197.6) \end{matrix})$	$(\begin{matrix} 0 \\ 0 \end{matrix})$	$(\begin{matrix} 0 \\ 0 \end{matrix})$
2.	$(\begin{matrix} (23815, 23815, 23815) \\ (22125, 22615, 23105) \end{matrix})$	$(\begin{matrix} (28557, 28576.2, 28595.4) \\ (23686.7, 24197.6, 24708.5) \end{matrix})$	$(\begin{matrix} 0 \\ 980 \end{matrix})$	$(\begin{matrix} 38.4 \\ 1021.7 \end{matrix})$
3.	$(\begin{matrix} (23815, 23815, 23815) \\ (23105, 22615, 22125) \end{matrix})$	$(\begin{matrix} (28557, 28576.2, 28595.4) \\ 23686.7, 24197.6, 24708.5 \end{matrix})$	$(\begin{matrix} 0 \\ - 980 \end{matrix})$	$(\begin{matrix} - 38.4 \\ - 1021.7 \end{matrix})$
4.	$(\begin{matrix} (23305, 23815, 24325) \\ (22615, 22615, 22615) \end{matrix})$	$(\begin{matrix} (27983.2, 28576.2, 29169.2) \\ (24184.3, 24197.6, 24210.9) \end{matrix})$	$(\begin{matrix} 1020 \\ 0 \end{matrix})$	$(\begin{matrix} 1186 \\ 26.5 \end{matrix})$
5.	$(\begin{matrix} (24325, 23815, 23305) \\ (22615, 22615, 22615) \end{matrix})$	$(\begin{matrix} (27983.2, 28576.2, 29169.2) \\ (24184.3, 24197.6, 24210.9) \end{matrix})$	$(\begin{matrix} - 1020 \\ 0 \end{matrix})$	$(\begin{matrix} - 1186 \\ - 26.5 \end{matrix})$
6.	$(\begin{matrix} (23305, 23815, 24325) \\ (22125, 22615, 23105) \end{matrix})$	$(\begin{matrix} (27964, 28576.2, 29188.4) \\ (23673.5, 23197.6, 24721.7) \end{matrix})$	$(\begin{matrix} 1020 \\ 980 \end{matrix})$	$(\begin{matrix} 1224.4 \\ 1048.3 \end{matrix})$
7.	$(\begin{matrix} (23305, 23815, 24325) \\ (23105, 22615, 22125) \end{matrix})$	$(\begin{matrix} (28002.4, 28576.2, 29150) \\ (24695.2, 24197.6, 23700) \end{matrix})$	$(\begin{matrix} 1020 \\ - 980 \end{matrix})$	$(\begin{matrix} 1147.5 \\ - 995.2 \end{matrix})$
8.	$(\begin{matrix} (24325, 23815, 23305) \\ (22125, 22615, 23105) \end{matrix})$	$(\begin{matrix} (29150, 28576.2, 28002.4) \\ (23700, 24197.6, 24695.2) \end{matrix})$	$(\begin{matrix} - 1020 \\ 980 \end{matrix})$	$(\begin{matrix} - 1147.5 \\ 995.2 \end{matrix})$
9.	$(\begin{matrix} (24325, 23815, 23305) \\ (23105, 22615, 22125) \end{matrix})$	$(\begin{matrix} (29188.4, 28576.2, 27964) \\ (24721.7, 24197.6, 23673.5) \end{matrix})$	$(\begin{matrix} - 1020 \\ - 980 \end{matrix})$	$(\begin{matrix} - 1224.4 \\ - 1048.3 \end{matrix})$

[i] Source: Author's own elaboration.

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DOI: https://doi.org/10.1515/ceej-2018-0017 | Journal eISSN: 2543-6821 | Journal ISSN: 2544-9001

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Language: English

Page range: 244 - 259

Published on: Sep 11, 2019

Published by: Faculty of Economic Sciences, University of Warsaw

In partnership with: Paradigm Publishing Services

Keywords:

Systems of linear equations,

interval arithmetic,

fuzzy numbers,

ordered fuzzy numbers,

Leontief model

JEL:

C02,

C30,

D57

Related subjects:

Business and economics,

Political economics,

Economic theory, systems and structures,

Microeconomics,

Macroeconomics,

Economic policy

© 2019 Dariusz Kacprzak, published by Faculty of Economic Sciences, University of Warsaw
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 5 (2018): Issue 52 (January 2018)

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

Figures & Tables

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Tab. 1

Fig. 5

Tab. 2

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