1 Introduction
2 Proposed approach
2.1 Bargaining game
2.2 Breakdown points in proposed model
2.3 Network game-DEA model
2.3.1 Game-DEA model for sub-network 1
2.3.2 Game-DEA model for sub-network 2
2.3.3 Game-DEA model for sub-network 3
2.3.4 Efficiency scores of whole network
2.4 Algorithm for assessing performance in network DEA
2.5 Assessing networks using the proposed model
3 A real case study of Iranian banking
Variable | Mean | Median | SD | Min | Max | Coef. Var. | Variable | Mean | Median | SD | Min | Max | Coef. Var. |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(x_{1j}^{11}\)
| 160.60 | 152.83 | 62.342 | 92.74 | 413.2 | 0.38 |
\(p_{2j}^{11}\)
| 97.37 | 97.31 | 10.40 | 81 | 127 | 0.107 |
\(x_{2j}^{11}\)
| 2851.2 | 2559 | 1635.3 | 614 | 7088 | 0.6 |
\(p_{1j}^{22}\)
| 43.04 | 38.52 | 30.28 | 3.85 | 104.16 | 0.704 |
\(x_{1j}^{12}\)
| 7.54 | 6.29 | 4.08 | 2.97 | 20.59 | 0.56 |
\(p_{2j}^{22}\)
| 99.65 | 83.9 | 76.25 | 4.62 | 294.6 | 0.77 |
\(x_{2j}^{12}\)
| 92.52 | 45.92 | 149.06 | 14.05 | 824.5 | 1.68 |
\(p_{1j}^{21}\)
| 3.31 | 2.8 | 0.99 | 2.8 | 7 | 0.30 |
\(x_{3j}^{12}\)
| 4.27 | 3.58 | 2.61 | 0.369 | 10.555 | 0.61 |
\(p_{2j}^{21}\)
| 18.73 | 18.65 | 3.78 | 11.67 | 28.89 | 0.20 |
\(x_{1j}^{21}\)
| 32.12 | 30.57 | 12.47 | 18.548 | 82.64 | 0.39 |
\(z_{1j}^{1}\)
| 25.7 | 22.5 | 13.08 | 11 | 60 | 0.51 |
\(x_{2j}^{21}\)
| 3421.4 | 3070.8 | 1962.3 | 736.8 | 8505.6 | 0.57 |
\(z_{2j}^{1}\)
| 2177.83 | 705.42 | 3440.3 | 113.75 | 16397.5 | 1.58 |
\(x_{1j}^{22}\)
| 9.26 | 7.71 | 5.16 | 2.97 | 24.21 | 0.56 |
\(z_{1j}^{2}\)
| 20.37 | 18.5 | 8.19 | 11 | 46 | 0.40 |
\(x_{2j}^{22}\)
| 28.76 | 6.67 | 67.33 | 0.752 | 371 | 2.34 |
\(z_{2j}^{2}\)
| 3711.2 | 1310.08 | 5141.76 | 211.25 | 20452.5 | 1.39 |
\(x_{3j}^{22}\)
| 0.98 | 0.67 | 1.01 | 0.059 | 4.75 | 1.03 |
\(y_{1j}\)
| 247.99 | 216.75 | 131.12 | 99 | 770.1 | 0.53 |
\(p_{1j}^{12}\)
| 79.91 | 46.16 | 79.44 | 1.98 | 294.6 | 0.99 |
\(y_{2j}\)
| 148.9 | 132 | 73.79 | 55 | 453 | 0.45 |
\(p_{2j}^{12}\)
| 375.94 | 262.4 | 339.56 | 11.55 | 1278.56 | 0.90 |
\(y_{3j}\)
| 268.83 | 254 | 115.56 | 110 | 683 | 0.43 |
\(p_{1j}^{11}\)
| 21.09 | 20.3 | 6.66 | 11.2 | 46.2 | 0.32 |
\(y_{4j}\)
| 60.84 | 58.9 | 10.24 | 44.49 | 86.83 | 0.17 |
4 Results and analysis
DMU | Sub-network 1 | Sub-network 2 | Sub-network 3 | Network | Classic DEA |
---|---|---|---|---|---|
1 | 1.0000000 (1) | 0.4978350 (9) | 0.5894470 (7) | 1.0000000 (1) | 1 (1) |
2 | 0.537162 (11) | 0.766234 (2) | 0.5096530 (8) | 0.8866660 (3) | 1 (1) |
3 | 0.277027 (26) | 0.244589 (22) | 1.0000000 (1) | 0.7911270 (6) | 1 (1) |
4 | 0.221284 (29) | 0.415584 (14) | 0.122265 (18) | 0.298579 (20) | 1 (1) |
5 | 0.461149 (16) | 0.393939 (15) | 0.377091 (10) | 0.621652 (11) | 1 (1) |
6 | 0.275338 (27) | 0.183983 (28) | 0.234234 (14) | 0.356035 (17) | 1 (1) |
7 | 0.491554 (12) | 0.056277 (30) | 0.086229 (20) | 0.185699 (27) | 1 (1) |
8 | 0.395270 (21) | 0.199134 (26) | 0.306306 (11) | 0.454563 (13) | 1 (1) |
9 | 0.395270 (21) | 0.484848 (10) | 0.4401540 (9) | 0.6806660 (9) | 1 (1) |
10 | 0.209459 (30) | 0.231602 (24) | 0.253539 (13) | 0.366429 (16) | 1 (1) |
11 | 0.366554 (23) | 0.5194810 (8) | 0.065637 (23) | 0.262427 (22) | 1 (1) |
12 | 0.6199320 (8) | 0.6839830 (4) | 0.019305 (30) | 0.173856 (28) | 1 (1) |
13 | 0.7618240 (4) | 0.7337660 (3) | 0.256113 (12) | 0.678540 (10) | 1 (1) |
14 | 0.454392 (17) | 0.465368 (11) | 0.175032 (15) | 0.439918 (14) | 1 (1) |
15 | 0.233108 (28) | 0.205628 (25) | 0.034749 (28) | 0.135250 (30) | 1 (1) |
16 | 0.366554 (23) | 0.196970 (27) | 0.8339770 (3) | 0.7340370 (8) | 1 (1) |
17 | 0.618243 (17) | 0.240260 (23) | 0.5920210 (6) | 0.7406870 (7) | 1 (1) |
18 | 0.601351 (10) | 0.6385280 (6) | 0.111969 (19) | 0.408444 (15) | 1 (1) |
19 | 0.396959 (20) | 0.461039 (12) | 0.6769630 (4) | 0.8344680 (4) | 1 (1) |
20 | 0.430743 (18) | 0.292208 (18) | 1.0000000 (1) | 0.9236000 (2) | 1 (1) |
21 | 0.364865 (25) | 0.264069 (20) | 0.028314 (29) | 0.145368 (29) | 1 (1) |
22 | 0.413851 (19) | 0.268398 (19) | 0.070785 (22) | 0.238168 (25) | 1 (1) |
23 | 0.8885140 (3) | 1.0000000 (1) | 0.155727 (16) | 0.594093 (12) | 1 (1) |
24 | 0.483108 (13) | 0.140693 (29) | 0.149292 (17) | 0.305917 (19) | 1 (1) |
25 | 0.472973 (14) | 0.5952380 (7) | 0.052767 (25) | 0.259459 (23) | 1 (1) |
26 | 0.6993240 (6) | 0.6709960 (5) | 0.037323 (27) | 0.247939 (24) | 1 (1) |
27 | 0.6435810 (7) | 0.337662 (17) | 0.064350 (24) | 0.268567 (21) | 1 (1) |
28 | 0.7043920 (5) | 0.448052 (13) | 0.084942 (21) | 0.338730 (18) | 1 (1) |
29 | 0.9121620 (2) | 0.246753 (21) | 0.041184 (26) | 0.216748 (26) | 1 (1) |
30 | 0.472973 (14) | 0.341991 (16) | 0.6705280 (4) | 0.8052440 (5) | 1 (1) |