Integrated dynamic interval data envelopment analysis in the presence of integer and negative data

Pooja Bansal; Aparna Mehra

doi:10.3934/jimo.2021023

Article Contents

2022, Volume 18, Issue 2: 1339-1363. Doi: 10.3934/jimo.2021023

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Integrated dynamic interval data envelopment analysis in the presence of integer and negative data

Pooja Bansal^, and
Aparna Mehra

Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016, India

^* Corresponding author: Pooja Bansal
^* Corresponding author: Pooja Bansal

Received: September 30, 2019

Revised: August 31, 2020

Early access: February 2021

Published: March 2022

The first author is supported by Council of Scientific Research (CSIR), India

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Abstract

The conventional data envelopment analysis (DEA) models presume that the values of input-output variables of the decision-making units (DMUs) are precisely known. However, some real-life situations can authoritatively mandate the data to vary in concrete fine-tuned ranges, which can include negative values and measures that are allowed to take integer values only. Our study proposes an integrated dynamic DEA model to accommodate interval-valued and integer-valued features that can take negative values. The proposed one-step model follows the directional distance function approach to determine the efficiency of DMUs over time in the presence of carryovers connecting the consecutive periods. We use the pessimistic and optimistic standpoints to evaluate the respective lower and upper bounds of the interval efficiency scores of the DMUs. We compare our proposed approach with a few relevant studies in the literature. We also validate our model on a synthetically generated dataset. Furthermore, we showcase the proposed procedure's applicability on a real dataset from 2014 to 2018 of airlines operating in India.

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Mathematics Subject Classification: Primary: 91B06, 65G30, 65G40, Secondary: 90C05, 90C08, 90C11, 90C39, 90C90.

Citation:

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Figure 1. The dynamic framework for a DMU comprising of $ T $ periods with multiple carryovers connecting consecutive periods

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Figure 2. 2a and 2b describe the upper and lower bounds respectively of efficient frontiers for DMUs $ 1 $ (red) and $ 3 $ (blue); the output is allowed to take integer values in the interval

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Figure 3. The dynamic structure of an airline in two consecutive periods

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Table 1. Results from models $ (M1) $ and $ (M2) $ on the data of five DMUs when $ \ell = \ell' = 1 $ and $ k = k' = 3 $; the output is restricted to take only integer values in the intervals

DMU$ _j $	$ [x_{1j}^L, \, x_{1j}^U] $	$ [y_{1j}^L, \, y_{1j}^U] $	$ E_j^{U} $	$ E_j^{L} $
1	[1,2]	[2,3]	1.0909	1.025
2	[2.5, 3]	[3,4]	1.0545	0.9405
3	[4,5]	[5,7]	1.5714	1.0435
4	[6,7]	[2,5]	1	0.76
5	[4, 4.5]	[1,3]	0.9643	0.8444

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Table 2. Results from models $ (M1) $ and $ (M2) $, $ \ell = \ell' = 1 $, $ k = k' = 3 $, in columns 2 and 3; and the interval SORM model [17] in columns 5 and 6 for the dataset of 20 banks described in [17]. Column 4 represents the classification of DMUs into three classes by our approach, while column 7 presents the classification into strictly efficient class ($ E^{++} $), weekly efficient class ($ E^{+} $), and inefficient class ($ E^{-} $) defined in [17]

DMU$ _j $	$ E_j^{L} $	$ E_j^{U} $	class (our approach)	$ \overline{E}_j $	$ \underline{E}_j $	class (by [17])
1	1.0063	1.0063	S	1	1	$ E^{++} $
2	0.9858	1.0116	E	0.813	1	$ E^{+} $
3	1.0006	1.0103	S	1	1	$ E^{++} $
4	0.9765	0.9889	IE	0.52	0.698	$ E^{-} $
5	1.0036	1.0084	S	1	1	$ E^{++} $
6	0.9678	0.9818	IE	0.427	0.614	$ E^{-} $
7	1.5072	1.5760	S	1	1	$ E^{++} $
8	0.9858	0.9896	IE	0.253	0.513	$ E^{-} $
9	1.0433	1.1944	S	1	1	$ E^{++} $
10	0.9616	0.9940	IE	0.718	0.901	$ E^{-} $
11	0.9984	1.0002	E	0.777	1	$ E^{+} $
12	0.9970	0.9990	IE	0.667	1	$ E^{+} $
13	0.9897	0.9939	IE	0.49	0.663	$ E^{-} $
14	0.9990	1.0042	E	0.929	1	$ E^{+} $
15	0.9940	1.0127	E	1	1	$ E^{++} $
16	0.9823	0.9928	IE	0.616	0.834	$ E^{-} $
17	0.9635	0.9823	IE	0.47	0.665	$ E^{-} $
18	0.9972	0.9990	IE	0.614	0.846	$ E^{-} $
19	0.9913	1.0051	E	0.866	1	$ E^{+} $
20	1.0066	1.0099	S	1	1	$ E^{++} $

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Table 3. Results from models $ (M1) $ and $ (M2) $ with necessary modifications, and the InDEA model in [26] on the data described in [26]. Here, $ Et^{L} $ and $ Et^{U} $ denote the lower and the upper bounds of the efficiency interval obtained by our models at time $ t $, while these bounds are represented by $ q^{Lt} $ and $ q^{Ut}, $ respectively, for the models in [26]. The symbols $ Q, \;Q^{+} $ and $ Q^{++} $ indicate the inefficient, efficient but not super-efficient, and super-efficient DMUs, respectively, by the classification defined in [26]. The ranking of units within each class is carried out by the PD method and recorded for both our proposed models and te one in [26]

DMU$ _j $ ($ j\to $)	1	2	3	4	5
$ E1^{L} $	1.0591	1.0275	0.9515	1.0108	0.9548
$ E1^{U} $	1.1077	1.0833	1.0065	1.0108	0.9730
Class	S	S	E	S	IE
Rank	1	2	4	3	5
$ q^{L1} $	1.03	1.06	0.74	0.70	0.62
$ q^{U1} $	2.02	2.04	1.04	0.87	0.85
Class	$ Q^{++} $	$ Q^{++} $	$ Q^{+} $	$ Q $	$ Q $
Rank	1	2	3	4	5
$ E2^{L} $	0.9527	1.0260	1.0559	1.0143	0.9547
$ E2^{U} $	1.0065	1.0773	1.1046	1.0143	0.9766
Class	E	S	S	S	IE
Rank	4	2	1	3	5
$ q^{L2} $	0.75	1.05	1.03	0.71	0.75
$ q^{U2} $	1.03	1.75	1.88	0.79	0.85
Class	$ Q^{+} $	$ Q^{++} $	$ Q^{++} $	$ Q $	$ Q $
Rank	3	2	1	5	4
$ E3^{L} $	1.0117	1.0291	0.9653	1.0675	0.9652
$ E3^{U} $	1.0682	1.1144	1.0151	1.1123	1.0220
Class	S	S	E	S	E
Rank	3	2	5	1	4
$ q^{L3} $	0.70	1.10	0.73	1.04	0.76
$ q^{U3} $	0.87	2.04	0.96	2.21	0.83
Class	$ Q $	$ Q^{++} $	$ Q $	$ Q^{++} $	$ Q $
Rank	5	2	3	1	4
$ E4^{L} $	1.0891	1.0818	0.9646	1.0225	0.9886
$ E4^{U} $	1.1149	1.0818	1.0236	1.0685	1.0330
Class	S	S	E	S	E
Rank	1	3	5	2	4
$ q^{L4} $	0.89	0.68	0.73	1.05	0.81
$ q^{U4} $	1.87	0.95	0.88	1.79	1.33
Class	$ Q^{+} $	$ Q $	$ Q $	$ Q^{++} $	$ Q^{+} $
Rank	2	4	1	5	3
$ E5^{L} $	0.9759	1.0442	0.9765	0.9424	0.9528
$ E5^{U} $	1.0569	1.0745	1.0185	0.9867	0.9839
Class	E	S	E	IE	IE
Rank	2	1	3	5	4
$ q^{L5} $	0.95	1.12	0.78	0.71	0.72
$ q^{U5} $	1.66	1.76	1.13	0.98	0.94
Class	$ Q^{+} $	$ Q^{++} $	$ Q^{+} $	$ Q $	$ Q $
Rank	2	1	3	4	5

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Table 4. Result from models $ (M1) $ and $ (M2) $ with $ \ell = \, \ell' = \, 1 $ and $ k = \, k' = \, 3 $, on the two periods synthetic dataset of 30 DMUs given in Appendix B

DMU$ _j $	interval of efficiency	class	rank
1	[0.7897, 1.2083]	E	28
2	[0.8055, 1.1376]	E	25
3	[0.8545, 1.3422]	E	6
4	[0.8891, 1.2386]	E	7
5	[0.8299, 1.2265]	E	21
6	[0.8856, 1.2316]	E	9
7	[0.8397, 1.2817]	E	15
8	[0.8396, 1.2332]	E	16
9	[0.8852, 1.224]	E	10
10	[0.7816, 1.2137]	E	29
11	[0.8582, 1.2269]	E	12
12	[0.7638, 1.1997]	E	30
13	[0.8442, 1.2267]	E	18
14	[0.8514, 1.1736]	E	13
15	[0.8054, 1.1402]	E	26
16	[0.8265, 1.2899]	E	17
17	[0.842, 1.1696]	E	19
18	[0.9258, 1.2041]	E	5
19	[0.8057, 1.2573]	E	23
20	[0.8351, 1.1974]	E	14
21	[0.8808, 1.2 759]	E	8
22	[0.8023, 1.2583]	E	24
23	[0.9288, 1.2615]	E	4
24	[0.8297, 1.2457]	E	22
25	[0.7959, 1.1546]	E	27
26	[0.875, 1.1831]	E	11
27	[0.9845, 1.2724]	E	3
28	[1.0122, 1.3216]	S	1
29	[0.9875, 1.1374]	E	2
30	[0.8106, 1.1771]	E	20

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Table 5. Inputs, outputs, desirable and undesirable carryovers applied in the empirical analysis

Inputs	Operating expenses (in millions INR)
Desirable carryover	Fleet size (in number)
Undesirable carryover	Losses carried forward after tax (in millions INR)
Outputs	Operating revenue (in millions INR)
	Pax load factor per month (in $ \% $)
	Weight load factor per month (in $ \% $)
	Passengers carried per month (in number)
	Cargo carried per month (in tonne)

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Table 6. Data of 11 Indian airlines for the period 2014-15

airline	fleet size	losses carried	operating expenses	operating revenue	pax load factor	weight load factor	passengers carried	cargo carried
1	101	59058.4	226854.4	206131.6	[73.7, 86.9]	[66.7, 77.6]	[990986,1217671]	[7940,10098]
2	17	611.7	19597.6	22948.2	[56.3, 93.8]	[45.4, 98.6]	[8430,18674]	[0, 32.8]
3	10	1789.2	3034	2279.5	[60.7, 72.2]	[55.3, 65.6]	[19612,38835]	[13,19]
4	3	1333.1	2885	1551.9	[71.7, 84]	[40.6, 53.2]	[68790,181483]	[450,1432]
5	5	-43.9	6310.4	6592	[0, 0]	[66.1, 71.9]	[0, 0]	[9542,11858]
6	19	-363.7	28715.8	30664.3	[75.6, 89.4]	[70.6, 82.4]	[534795,639264]	[3932,5485]
7	94	-16590.3	123578.6	139253.4	[76.8, 91.9]	[68.1, 86.4]	[2230645,2769283]	[10300, 12303.8]
8	107	18137.1	215030.1	195606.1	[77.1, 89.5]	[69.7, 78.6]	[1186492,1393452]	[6355,9124]
9	9	2876.5	16775.2	14229.4	[74.9, 89.7]	[68.8, 79.8]	[171551,294766]	[915,1385]
10	34	6870.5	60885	52015.3	[80, 93.4]	[75.3, 87.7]	[552726,976517]	[3202,6208]
11	6	1990.7	2681.9	691.3	[45.4, 77.6]	[34.9, 66.6]	[14999,158348]	[0, 2151]

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Table 7. Results of the dynamic integrated interval efficiency models $ (M1) $ and $ (M2) $ with $ \ell = \ell' = 1 $ and $ k = k' = 3, $ and ranking of super-efficient and efficient DMUs

			Super-efficient DMUs		Efficient DMUs
airline	efficiency interval	class	ranking vector	rank	ranking vector	rank
1	[1.3333, 1.3938]	S	7.5	2
2	[1.0026, 1.1066]	S	2.459	7
3	[1.0156, 1.0369]	S	1.356	9
4	[1.0052, 1.2314]	S	4.053	6
5	[1.1249, 1.2199]	S	5.373	3
6	[0.9994, 1.0895]	E			1.051	10
7	[1.4685, 1.6575]	S	8.5	1
8	[1.0950, 1.1924]	S	4.847	4
9	[0.9986, 1.0734]	E			0.949	11
10	[1.0210, 1.2842]	S	4.630	5
11	[1.0081, 1.0591]	S	1.781	8

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Table 8. Synthetic dataset of 30 DMUs for $ t = 1 $ with two inputs, two outputs, and one desirable carryover linking the two periods

DMU$ _j $	$ X_{1} $	$ X_{2} $	$ Y_{1} $	$ Y_{2} $	$ C_{1} $
1	[0, 3]	[-0.1615, 3.0767]	[2,10]	[3.6768, 7.233]	[-0.9007, 0.3911]
2	[-5, 0]	[-1.182, 7.4288]	[1,3]	[5.0074, 7.5827]	[-3.833, -1.0627]
3	[-2, -1]	[2.863, 6.6322]	[-2, 6]	[6.5486, 11.8582]	[0.7328, 7.2357]
4	[-11, -5]	[5.5777, 9.1795]	[-4, -3]	[7.2685, 14.1149]	[3.8294, -2.4358]
5	[-3, 1]	[0.6484, 5.7141]	[1,8]	[6.5704, 9.7772]	[3.5951, 3.4671]
6	[3,6]	[-0.2489, 4.7478]	[10,10]	[5.5064, 9.1181]	[5.1994, 0.0173]
7	[-6, -2]	[-5.4287, -2.6282]	[0, 3]	[-1.6806, 3.1973]	[-6.5508, -0.0203]
8	[-2, 2]	[3.4445, 5.0916]	[1,8]	[6.8983, 9.4379]	[-0.7025, -3.7166]
9	[2,5]	[-5.9021, -2.2483]	[7,12]	[-1.7842, 0.9635]	[3.2851, -1.2266]
10	[-2, 3]	[-0.7791, 2.458]	[2,7]	[3.2602, 5.8106]	[-3.2601, -0.8168]
11	[-9, -6]	[4.5724, 8.613]	[-2, -1]	[10.5083, 15.6547]	[-2.0706, 1.0867]
12	[-1, 2]	[1.582, 4.8292]	[2,4]	[4.7475, 7.3799]	[-5.0713, -2.4425]
13	[1,3]	[5.5856, 5.9474]	[5,8]	[9.8552, 9.9694]	[-5.5407, 0.0637]
14	[-6, -4]	[-1.6635, 1.3701]	[2,2]	[1.7472, 2.1861]	[-2.9329, -1.6078]
15	[-3, 1]	[-3.975, -1.1701]	[3,3]	[-0.7801, 0.2799]	[-2.3205, -1.5946]
16	[-7, 0]	[-6.6192, -0.3315]	[-4, 0]	[-1.9427, -1.3506]	[-6.3538, -0.7493]
17	[-4, 0]	[1.1301, 5.0351]	[0, 0]	[6.6786, 8.0126]	[-1.7038, 1.2584]
18	[6,8]	[-3.3653, 2.1456]	[12,13]	[1.6537, 3.4448]	[-1.2121, 5.5107]
19	[-5, 0]	[6.2833, 7.7517]	[-2, 0]	[8.1914, 12.7517]	[0.4046, 3.4051]
20	[-3, -1]	[1.1149, 1.1633]	[-2, 3]	[5.6438, 8.1306]	[-1.0965, 5.2077]
21	[-5, 4]	[8.9549, 9.3198]	[5,8]	[11.0651, 17.3472]	[-0.981, -5.7919]
22	[3,5]	[-3.7758, -0.397]	[4,7]	[2.2879, 3.0219]	[-1.1107, -3.3136]
23	[-8, -1]	[-5.5734, -1.605]	[-1, 3]	[-2.0838, 4.5846]	[4.0279, -0.1544]
24	[-2, 4]	[-2.359, 1.1816]	[2,5]	[0.7777, 3.9289]	[-0.2559, -3.4568]
25	[-1, 5]	[2.6094, 2.7898]	[5,6]	[3.9727, 6.8607]	[-0.5585, 0.7262]
26	[-1, -1]	[2.9899, 6.6723]	[5,6]	[8.6948, 11.5419]	[-1.5396, -3.3929]
27	[0, 5]	[-3.0471, 3.6269]	[4,9]	[1.2691, 4.5094]	[5.9166, 1.5114]
28	[0, 5]	[-2.8085, 4.2057]	[2,11]	[4.4506, 5.2129]	[2.597, 9.379]
29	[-1, 3]	[1.7709, 3.0374]	[2,6]	[5.6242, 6.2396]	[7.127, -3.5507]
30	[-5, -2]	[-1.5041, 1.8533]	[-2, 4]	[2.1227, 5.3259]	[-1.9647, -4.8179]

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Table 9. Synthetic dataset of 30 DMUs for $ t = 2 $ with two inputs, two outputs

DMU$ _j $	$ X_{1} $	$ X_{2} $	$ Y_{1} $	$ Y_{2} $
1	[-3, 1]	[1,3]	[-11.36, -3.2555]	[-3.3936, -1.5096]
2	[1,1]	[2,5]	[-0.9634, 1.7725]	[1.7725, 4.0171]
3	[-7, -1]	[0, 2]	[1.8694, 3.4901]	[3.4901, 8.0933]
4	[-3, 7]	[5,7]	[-0.4754, 4.3661]	[4.3661, 2.8741]
5	[-3, 2]	[0, 4]	[-2.6313, 4.1259]	[4.1259, 2.8374]
6	[-5, 0]	[1,3]	[0.6498, 3.7561]	[3.7561, 2.7529]
7	[-1, 6]	[4,9]	[-9.4604, -4.8163]	[-4.8163, -5.7451]
8	[-8, -4]	[-4, -1]	[-1.7383, -0.5501]	[-0.5501, 1.5019]
9	[2,5]	[6,10]	[6.2155, 9.9875]	[9.9875, 11.2043]
10	[-4, 4]	[6,6]	[0.3967, 3.3566]	[3.3566, 4.4534]
11	[-3, -1]	[-2, -2]	[-3.638, -0.2749]	[-0.2749, 2.2454]
12	[-1, 8]	[4,11]	[-6.8705, -3.8813]	[-3.8813, -2.2912]
13	[-6, 2]	[1,4]	[1.3005, 3.6518]	[3.6518, 5.8577]
14	[0, 1]	[1,5]	[4.6112, 7.6644]	[7.6644, 7.6551]
15	[3,5]	[3,8]	[-3.1301, -0.6347]	[-0.6347, 0.9748]
16	[-6, -3]	[-1, 2]	[-1.3322, 0.3764]	[0.3764, 0.6187]
17	[-3, -2]	[0, 4]	[-8.6088, -6.2055]	[-6.2055, -2.6939]
18	[1,5]	[5,8]	[-1.0369, 5.3176]	[5.3176, 5.2215]
19	[1,4]	[6,12]	[-3.0645, 2.224]	[2.224, 0.3195]
20	[-4, 0]	[1,5]	[-5.066, -1.4253]	[-1.4253, -1.7729]
21	[-2, 3]	[3,3]	[4.9521, 8.3288]	[8.3288, 7.6476]
22	[2,4]	[3,15]	[-1.5163, 3.1201]	[3.1201, 4.2655]
23	[-5, -2]	[-2, 1]	[-5.5681, -1.3893]	[-1.3893, -1.5713]
24	[-10, -4]	[-4, -2]	[-4.6455, 1.4948]	[1.4948, 1.8267]
25	[-3, 2]	[0, 4]	[-3.4395, 6.5853]	[6.5853, 7.1234]
26	[2,6]	[8,9]	[7.6897, 9.6555]	[9.6555, 10.8196]
27	[2,3]	[6,12]	[-7.7017, -1.5092]	[-1.5092, -1.0977]
28	[2,8]	[8,10]	[8.3769, 11.9711]	[11.9711, 13.669]
29	[-1, 5]	[0, 7]	[-6.1897, 0.5822]	[0.5822, 0.6056]
30	[-3, 3]	[3,4]	[-1.7567, 0.7063]	[0.7063, 2.1542]

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