A slacks-based model for dynamic data envelopment analysis

Mohammad Afzalinejad; Zahra Abbasi

doi:10.3934/jimo.2018043

Article Contents

2019, Volume 15, Issue 1: 275-291. Doi: 10.3934/jimo.2018043

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A slacks-based model for dynamic data envelopment analysis

Mohammad Afzalinejad^, and
Zahra Abbasi

Department of Mathematics, Tafresh University, Tafresh, 3951879611, Iran

* Corresponding author: Mohammad Afzalinejad
* Corresponding author: Mohammad Afzalinejad

Received: April 30, 2017

Revised: September 30, 2017

Early access: April 2018

Published: December 2018

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Abstract

Dynamic Data Envelopment Analysis (DDEA) deals with efficiency analysis of decision making units in time dependent situations. A finite number of time periods and some carry-over activities between each two consecutive periods are assumed in DDEA. There are many models in DEA for efficiency evaluation of decision making units over time periods. One important class of dynamic models is the class of slacks-based models. By using a numerical example we show that some slacks-based DDEA models, especially ones proposed by Tone and Tsutsui, suffer from efficiency overestimation. A new dynamic slacks-based DEA model is proposed to overcome the deficiencies of the available slacks-based models. The model proposed in this paper is capable of revealing all sources of inefficiencies and providing more discrimination between decision making units. The theoretical and practical examinations demonstrate the merits of the new model.

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Mathematics Subject Classification: Primary: 90C05, 90B50; Secondary: 91B06.

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Figure 1. $DMU_{b}$ is dominated by $DMU_{a}$ and is not efficient.

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Figure 2. Comparison of rank order (input-oriented).

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Table 1. Data of 10 bank branches

DMUs	Average monthly salaries			Operating expense			Total loans			Net profit			Loan Losses
DMUs	t=1	t=2	t=3	t=1	t=2	t=3	t=1	t=2	t=3	t=1	t=2	t=3	t=1	t=2	t=3
DMU1	2.828	2.705	3.775	27.55	35.25	50.43	40.01	49.85	54.38	57.95	58.85	66.64	12.41	7.88	7.4
DMU2	5.667	5.825	7.657	84.5	122	105.5	282.9	297.6	322.5	94.18	87.29	111.6	41.34	34.95	28.64
DMU3	6.23	6.32	8.899	183.6	159.5	170.8	184.5	191.4	188.4	103.7	120.5	121.6	28.44	22.71	21.41
DMU4	5.577	5.532	7.552	122.7	94.48	94.97	195.9	200.5	202.2	58.98	58.42	58.25	22.8	25.68	26.69
DMU5	3.864	4.526	5.72	57.19	38.43	40.27	106.2	102.9	98.36	32.41	42.5	48.91	8.51	6.25	8.93
DMU6	4.696	4.601	6.196	72.07	2.64	3.41	175.5	176.1	190.7	60.7	58.88	47.68	10.35	11.89	10.22
DMU7	3.582	3.108	4.221	21.83	21.3	29.76	21.56	24.38	28.28	18.68	19.17	19.42	1.91	1.24	2.02
DMU8	5.395	5.522	7.139	63.85	56.14	49	133	147.1	156.8	76.77	99.79	100.9	30.49	21.06	18.07
DMU9	7.761	7.522	10.746	27.93	34.4	31.14	872.9	815.4	803.3	314.7	312.8	31.21	80.96	119.5	115.5
DMU10	3.748	3.593	5.138	59.99	96.5	60.43	113.7	121.6	122.9	72.64	84.51	81.45	7.33	3.28	13.53

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Table 2. Comparison of the efficiency scores resulted from Tone and Tsutsui's model and the proposed model.

	Overall input-oriented efficiency		Overall output-oriented efficiency			Non-oriented combined efficiency
DMUs	$ \theta^*_{o}(\text{TT})$	$ \theta^{*}_{o}$	$ \tau^{*}_{o}(\text{TT})$	$ \varphi^{*}_{o}$	Model (16) with TT objective: $ \varphi^{*}_{o}(\text{TT})$	$ \theta^{}_{o}(\text{TT})\times\tau^{}_{o}(TT)$	$ \theta^{}_{o}\times\varphi^{}_{o}$
DMU1	0.9194	0.8792	0.6771	0.7587	0.6771	0.6225	0.6671
DMU2	1	0.7040	1	0.8458	0.7492	1	0.5954
DMU3	0.6521	0.4735	0.7618	0.7959	0.7230	0.4968	0.3761
DMU4	0.5133	0.6773	0.5840	0.7117	0.5456	0.2998	0.4821
DMU5	0.7648	0.7614	0.7916	0.8100	0.7286	0.6054	0.6168
DMU6	1	1	1	1	1	1	1
DMU7	0.8854	0.6421	0.9409	0.8979	0.8700	0.8331	0.5766
DMU8	0.7765	0.7020	0.7482	0.7766	0.6841	0.5810	0.5451
DMU9	1	1	1	1	1	1	1
DMU10	1	0.5660	0.1	0.6979	0.9977	1	0.3950

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Table 3. The input-oriented period efficiency scores resulted from the TT model and the proposed model

DMUs	Tone and Tsutsui's model			The proposed model
DMUs	Period 1 efficiency	Period 2 efficiency	Period 3 efficiency	Period 1 efficiency	Period 2 efficiency	Period 3 efficiency
DMU1	0.7574	1	1	0.6364	1	1
DMU2	1	1	1	0.2769	0.8169	1
DMU3	0.5217	0.6987	0.7450	0.1962	0.4393	0.7247
DMU4	0.3887	0.6334	0.5324	0.1802	0.8321	1
DMU5	0.7047	0.8411	0.7477	0.5053	0.7744	1
DMU6	1	1	1	1	1	1
DMU7	0.6562	1	1	0.4236	0.5221	0.9991
DMU8	0.4524	0.8718	1	0.2319	0.8498	1
DMU9	1	1	1	1	1	1
DMU10	1	1	1	0.2602	0.8350	0.5901

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