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

• * Corresponding author: Pooja Bansal

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

• 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.

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

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

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

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

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

Table 2.  Results from models $(M1)$ and $(M2)$, $\ell = \ell' = 1$, $k = k' = 3$, in columns 2 and 3; and the interval SORM model  in columns 5 and 6 for the dataset of 20 banks described in . 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 

 DMU$_j$ $E_j^{L}$ $E_j^{U}$ class (our approach) $\overline{E}_j$ $\underline{E}_j$ class (by ) 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 $E^{++}$

Table 3.  Results from models $(M1)$ and $(M2)$ with necessary modifications, and the InDEA model in  on the data described in . 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 . The symbols $Q, \;Q^{+}$ and $Q^{++}$ indicate the inefficient, efficient but not super-efficient, and super-efficient DMUs, respectively, by the classification defined in . 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 

 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

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

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)

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]

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

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]

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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