REDISTRIBUTING MULTIPLE INPUTS WITH A PARABOLIC DEA MODEL

Moreira, Luíza Serra; Soares de Mello, João Carlos C. B.; Angulo Meza, Lidia

doi:10.1590/0101-7438.2023.043.00267995

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ARTICLES • Pesqui. Oper. 43 • 2023 • https://doi.org/10.1590/0101-7438.2023.043.00267995 copy

REDISTRIBUTING MULTIPLE INPUTS WITH A PARABOLIC DEA MODEL

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

Parametric DEA models presume a functional form for the efficiency frontier and are used for resource redistribution. Parabolic DEA is a particular case of these models used to redistribute a single input in Variable Returns to Scale scenarios. In this study, we extend this model to redistribute multiple inputs simultaneously, ensuring that all DMUs will become extreme efficient after redistribution maintaining their outputs. The proposed model is a single multi-objective Linear Programming Problem (PPL) which provides a single optimal solution. To solve this model, two approaches are used, the Weighted Sum of the Objective Functions and the Separation of Variables. Two numerical examples considering single and multiple outputs are used and the results obtained are identical for the two approaches.

Keywords:
parabolic DEA; resources redistribution; variable returns to scale

DMU	x ₁	x ₂	y	BCC eff.
A	0.90	6.00	1.00	1.00
B	4.10	5.00	2.00	1.00
C	8.80	8.00	2.50	0.82
D	16.00	10.00	3.50	0.84
E	25.20	9.00	5.40	1.00
Total	55.00	38.00	14.40

DMU	x ₁	x ₂	y	BCC eff.
A	0.04	0.60	0.19	1.00
B	0.16	0.50	0.37	1.00
C	0.35	0.80	0.46	0.82
D	0.63	1.00	0.65	0.84
E	1.00	0.90	1.00	1.00
Total	2.18	3.80	2.67

DMU	Result 1		Result 2		Result 3
	λ₁=0.5	λ₂=0.5	λ₁=0.9	λ₂=0.1	λ₁=0.3	λ₂=0.7
	x₁	x₂	x₁	x₂	x₁	x₁
A	0.12	0.68	0.12	0.68	0.12	0.68
B	0.24	0.71	0.24	0.71	0.24	0.71
C	0.31	0.73	0.31	0.73	0.31	0.73
D	0.51	0.78	0.51	0.78	0.51	0.78
E	1.00	0.90	1.00	0.90	1.00	0.90
Total	2.18	3.80	2.18	3.80	2.18	3.80

DMU	LP 1	LP 2
DMU	x ₁	x ₂
A	0.12	0.68
B	0.24	0.71
C	0.31	0.73
D	0.51	0.78
E	1.00	0.90
Total	2.18	3.80

DMU	Original data		Resolution 1		Resolution 2		Efficiency BCC
DMU	x ₁	x ₂	x ₁	x ₂	x ₁	x ₂	Original	Final
A	0.04	0.60	0.12	0.68	0.12	0.68	1.00	1.00
B	0.16	0.50	0.24	0.71	0.24	0.71	1.00	1.00
C	0.35	0.80	0.31	0.73	0.31	0.73	0.82	1.00
D	0.63	1.00	0.51	0.78	0.51	0.78	0.84	1.00
E	1.00	0.90	1.00	0.90	1.00	0.90	1.00	1.00
Total	2.18	3.80	2.18	3.80	2.18	3.80	2.18	3.80

DMU	x ₁	x ₂	y ₁	y ₂	BCC ef.
A	0.90	6.00	1.00	4.00	1.00
B	4.10	5.00	2.00	3.00	1.00
C	8.80	8.00	2.50	8.00	1.00
D	16.00	10.00	3.50	6.00	0.84
E	25.20	9.00	5.40	10.00	1.00
Total	55.00	38.00	14.40	31.00

DMU	x ₁	x ₂	y ₁	y ₂
A	0.04	0.60	0.19	0.40
B	0.16	0.50	0.37	0.30
C	0.35	0.80	0.46	0.80
D	0.63	1.00	0.65	0.60
E	1.00	0.90	1.00	1.00
Total	2.18	3.80	2.67	3.10

DMU	Results 1		Results 2		Results 3
	λ₁ =0.5	λ₂ =0.5	λ₁ =0.1	λ₂ =0.9	λ₁ =0.7	λ₂ =0.3
	x ₁	x ₂	x ₁	x ₂	x ₁	x ₂
A	0.11	0.60	0.11	0.60	0.11	0.60
B	0.20	0.60	0.20	0.60	0.20	0.60
C	0.35	0.80	0.35	0.80	0.35	0.80
D	0.49	0.75	0.49	0.75	0.49	0.75
E	1.03	1.05	1.03	1.05	1.03	1.05

DMU	PPL 1	PPL 2
DMU	x ₁	x ₂
A	0.11	0.60
B	0.20	0.60
C	0.35	0.80
D	0.49	0.75
E	1.03	1.05

DMU	Original Data		Resolution 1		Resolution 2		Efficiency BCC
DMU	x ₁	x ₂	x ₁	x ₂	x ₁	x ₂	Original	Final
A	0.04	0.60	0.11	0.60	0.11	0.60	1.00	1.00
B	0.16	0.50	0.20	0.60	0.20	0.60	1.00	1.00
C	0.35	0.80	0.35	0.80	0.35	0.80	1.00	1.00
D	0.63	1.00	0.49	0.75	0.49	0.75	0.84	1.00
E	1.00	0.90	1.03	1.05	1.03	1.05	1.00	1.00

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