Introduction
Literature review
Motivations and contributions
Organization
Relational NDEA model
\(n\) | Number of DMUs |
\(m\) | Number of inputs of system |
\(s\) | Number of outputs of system |
\(X^{(t)}\) | Inputs of process \(t\) |
\(Y^{(t)}\) | Outputs of process \(t\) |
\(y_{{r_{t} j}}^{(t)(O)}\) | Final outputs of system produced by process \(t\) |
\(y_{{r_{t} j}}^{(t)(I)}\) | Intermediate outputs of process \(t\) consumed by process 3 |
\(v\) | Weight vector of inputs |
\(u^{(t)}\) | Weight vector of outputs of process \(t\) |
\(E_{{_{K} }}\) | System efficiency of \({\text{DMU}}_{k}\) |
\(E_{k}^{(t)}\) | Efficiency of \({\text{DMU}}_{k}\) for process \(t\) |
\(E_{k}^{t}\) | Efficiency of \({\text{DMU}}_{k}\) for stage \(t\) |
\(u^{(t)L}\) | Absolute lower bounds imposed on output weights of process \(t\) |
\(u^{(t)U}\) | Absolute upper bounds imposed on output weights of process \(t\) |
\(v^{L}\) | Absolute lower bounds imposed on input weights |
\(v^{U}\) | Absolute upper bounds imposed on input weights |
\(\Omega\) | Deviation variable |
\(\mathop {}\limits^{ \sim }\) | Fuzzy notation |
Fuzzy relational NDEA model
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Triangular and trapezoidal membership functions are most often used for representing fuzzy numbers in real-life applications of fuzzy DEA models [25, 34, 42, 45, 46]. The main reason for using triangular or trapezoidal fuzzy numbers is that they are easy to use and interpret. However, other shapes, such as the power membership function and Gaussian membership function are preferable in some applications. Triangular, trapezoidal, power, and Gaussian fuzzy numbers are all LR flat fuzzy numbers [17, 18]. Since the proposed approach in this research is based on alpha cut of fuzzy data, in contrast to some existing fuzzy DEA approaches, it is applicable for solving fuzzy DEA models involving any kind of LR flat fuzzy numbers.
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In contrast to fuzzy arithmetic approach [65] that (1) is limited for solving DEA models with triangular fuzzy numbers and (2) ignores the internal structure of production process of a DMU and focus on a single process, the proposed approach (1) can be applied for solving fuzzy relational network DEA models with any kind of flat LR fuzzy numbers, (2) enables us to deal with the mixed network structures, and (3) allows us to consider equality of opportunity in a fuzzy environment when evaluating the system efficiency and the component process efficiencies.
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Kao and Liu [47] used the extension principle to find the lower and upper bounds of fuzzy efficiency of each DMU in a fuzzy two-stage DEA model. To do this, they solved two distinct mathematical programming models, one liner model for calculating the upper bound and one non-linear model for calculating the lower bound, for each DMU at given alpha level. Hence, that approach needs to solve \(2n\alpha\) models to find the interval efficiency of all DMUs. In contrast to this approach that provides interval efficiency for each DMU and needs an additional effort for complete ranking of all DMUs, our proposed approach gives a crisp efficiency for each DMU at given alpha level. In contrast to Kao and Liu’s [47] approach, the proposed approach in this study solves linear programming models for calculating the fuzzy efficiency of each DMU. Moreover, the proposed fuzzy relational network approach provides the efficiency of all DMUs by solving a single linear model at given alpha level. Finally, the present approach equitably evaluates the efficiencies of the system and their processes for all DMUs by assigning common set of weights.
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Hatami-Marbini and Saati [38] used the alpha cut approach to find the lower and upper bounds of fuzzy efficiency of each DMU in a fuzzy two-stage DEA model by finding common set of weights. In contrast to Hatami-Marbini and Saati’s [38] approach, the proposed approach in this study enables us to deal with fuzzy mixed network DEA models. Moreover, the common-weights approach proposed by Hatami-Marbini and Saati [38] provides interval efficiencies for all DMUs. However, their proposed formulations to obtain the lower and upper bounds of efficiency measures do not provide the efficiency scores within the interval (0, 1]. Thus, the efficiency values are out of the range (0, 1]. In contrast to that approach, the proposed fuzzy relational network approach not only provide a unique efficiency instead of interval one, but also provides the efficiency measures within the range of (0, 1].
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In contrast to defuzzification approach [66] that ignores the uncertainty in inputs and outputs, the alpha cut approach concerts effort to preserve the fuzzy information without detriment of the decision maker's intuition and subjective judgments in the performance assessment.
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In contrast to fuzzy ranking approach [34] that ignores the possible range of fuzzy efficiency at given alpha level and requires solving a bi-level linear programming model, the proposed approach requires solving a linear programming problem with less computational effort.
Numerical example
DMU | System/process | |||||||
---|---|---|---|---|---|---|---|---|
A | System | (8, 11, 17) | (11, 14, 20) | (1, 2, 3) | – | (1, 2, 3) | – | (0, 1, 2) |
Process 1 | (2, 3, 5) | (4, 5, 7) | (1, 2, 3) | (1, 2, 3) | – | – | – | |
Process 2 | (3, 4, 6) | (2, 3, 5) | – | – | (1, 2, 3) | (0, 1, 2) | – | |
Process 3 | (3, 4, 6) | (5, 6, 8) | – | (1, 2, 3) | – | (0, 1, 2) | (0, 1, 2) | |
B | System | (4, 7, 11) | (4, 7, 12) | (0, 1, 2) | – | (0, 1, 2) | – | (0, 1, 2) |
Process 1 | (1, 2, 3) | (2, 3, 5) | (0, 1, 2) | (0, 1, 2) | – | – | – | |
Process 2 | (1, 2, 3) | (0, 1, 2) | – | – | (0, 1, 2) | (0, 1, 2) | – | |
Process 3 | (2, 3, 5) | (2, 3, 5) | – | (0, 1, 2) | – | (0, 1, 2) | (0, 1, 2) | |
C | System | (8, 11, 17) | (11, 14, 20) | (0, 1, 2) | – | (0, 1, 2) | – | (1, 2, 3) |
Process 1 | (2, 3, 5) | (3, 4, 6) | (0, 1, 2) | (0, 1, 2) | – | – | – | |
Process 2 | (4, 5, 7) | (2, 3, 5) | – | – | (0, 1, 2) | (0, 1, 2) | – | |
Process 3 | (2, 3, 5) | (6, 7, 9) | – | (0, 1, 2) | – | (0, 1, 2) | (1, 2, 3) | |
D | System | (11, 14, 20) | (11, 14, 20) | (1, 2, 3) | – | (2, 3, 5) | – | (0, 1, 2) |
Process 1 | (3, 4, 6) | (5, 6, 8) | (1, 2, 3) | (0, 1, 2) | – | – | – | |
Process 2 | (4, 5, 7) | (4, 5, 7) | – | – | (2, 3, 5) | (0, 1, 2) | – | |
Process 3 | (4, 5, 7) | (2, 3, 5) | – | (0, 1, 2) | – | (0, 1, 2) | (0, 1, 2) | |
E | System | (11, 14, 20) | (12, 15, 21) | (2, 3, 5) | – | (1, 2, 3) | – | (2, 3, 5) |
Process 1 | (4, 5, 7) | (5, 6, 8) | (2, 3, 5) | (0, 1, 2) | – | – | – | |
Process 2 | (4, 5, 7) | (3, 4, 6) | – | – | (1, 2, 3) | (1, 2, 3) | – | |
Process 3 | (3, 4, 6) | (4, 5, 7) | – | (0, 1, 2) | – | (1, 2, 3) | (2, 3, 5) |
UB | \(\alpha\) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
0.0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | |
\(\hat{v}_{1}\) | 0.09 | 0.09 | 0.09 | 0.08 | 0.08 | 0.08 | 0.08 | 0.08 | 0.07 | 0.07 | 0.07 |
\(\hat{v}_{2}\) | 0.08 | 0.08 | 0.08 | 0.08 | 0.08 | 0.07 | 0.07 | 0.07 | 0.07 | 0.07 | 0.07 |
\(\hat{u}_{1}^{(1)}\) | 0.2917 | 0.2513 | 0.2182 | 0.1908 | 0.1677 | 0.1481 | 0.1313 | 0.1168 | 0.1042 | 0.0931 | 0.0833 |
\(\hat{u}_{1}^{(2)}\) | 0.3182 | 0.2735 | 0.2371 | 0.2068 | 0.1815 | 0.1600 | 0.1416 | 0.1254 | 0.1026 | 0.0852 | 0.0714 |
\(\hat{u}_{1}^{(3)}\) | 1.3958 | 1.1336 | 0.9301 | 0.7697 | 0.6417 | 0.5270 | 0.4316 | 0.3589 | 0.3013 | 0.2598 | 0.2167 |
UB | \(\alpha\) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
0.0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | |
\(v_{1}^{**}\) | 0.0315 | 0.0345 | 0.0342 | 0.0322 | 0.0364 | 0.0400 | 0.0400 | 0.0393 | 0.0350 | 0.0350 | 0.0345 |
\(v_{2}^{**}\) | 0.0216 | 0.0345 | 0.0284 | 0.0322 | 0.0364 | 0.0350 | 0.0350 | 0.0344 | 0.0350 | 0.0350 | 0.0345 |
\(u_{1}^{(1)**}\) | 0.0784 | 0.0394 | 0.0781 | 0.0766 | 0.0774 | 0.0750 | 0.0650 | 0.0610 | 0.0500 | 0.0450 | 0.0394 |
\(u_{1}^{(2)**}\) | 0.0865 | 0.0345 | 0.0852 | 0.0846 | 0.0820 | 0.0800 | 0.0700 | 0.0661 | 0.0500 | 0.0450 | 0.0345 |
\(u_{1}^{(3)**}\) | 0.3784 | 0.0116 | 0.3300 | 0.3104 | 0.2914 | 0.2650 | 0.2150 | 0.1769 | 0.1500 | 0.1250 | 0.1116 |
OSI | \(\alpha\) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
0.0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | |
\(\overline{x}_{11}^{**}\) | 0.2523 | 0.3793 | 0.2938 | 0.2870 | 0.3352 | 0.3800 | 0.3920 | 0.3971 | 0.3640 | 0.3745 | 0.3793 |
\(\overline{x}_{12}^{**}\) | 0.2378 | 0.4828 | 0.3293 | 0.3837 | 0.4444 | 0.4375 | 0.4480 | 0.4506 | 0.4690 | 0.4795 | 0.4828 |
\(\overline{x}_{21}^{**}\) | 0.1261 | 0.2414 | 0.1572 | 0.1580 | 0.1899 | 0.2220 | 0.2320 | 0.2398 | 0.2240 | 0.2345 | 0.2414 |
\(\overline{x}_{22}^{**}\) | 0.0865 | 0.2414 | 0.1306 | 0.1580 | 0.1894 | 0.1925 | 0.2030 | 0.2098 | 0.2240 | 0.2345 | 0.2414 |
\(\overline{x}_{31}^{**}\) | 0.2523 | 0.3793 | 0.2938 | 0.2870 | 0.3352 | 0.3800 | 0.3920 | 0.3971 | 0.3640 | 0.3745 | 0.3793 |
\(\overline{x}_{32}^{**}\) | 0.2378 | 0.4828 | 0.3293 | 0.3837 | 0.4444 | 0.4375 | 0.4480 | 0.4506 | 0.4690 | 0.4795 | 0.4828 |
\(\overline{x}_{41}^{**}\) | 0.3468 | 0.4828 | 0.3963 | 0.3837 | 0.4444 | 0.5000 | 0.5120 | 0.5150 | 0.4690 | 0.4795 | 0.4828 |
\(\overline{x}_{42}^{**}\) | 0.2378 | 0.4828 | 0.2938 | 0.3837 | 0.4444 | 0.4375 | 0.4480 | 0.4506 | 0.4690 | 0.4795 | 0.4828 |
\(\overline{x}_{51}^{**}\) | 0.6306 | 0.4828 | 0.3293 | 0.5840 | 0.5191 | 0.5000 | 0.5120 | 0.5150 | 0.4690 | 0.4795 | 0.4828 |
\(\overline{x}_{52}^{**}\) | 0.3694 | 0.5172 | 0.3963 | 0.4160 | 0.4809 | 0.4725 | 0.4830 | 0.4850 | 0.5040 | 0.5145 | 0.5172 |
\(\overline{x}_{11}^{(1)**}\) | 0.1577 | 0.1712 | 0.3646 | 0.1419 | 0.1530 | 0.1600 | 0.1520 | 0.1415 | 0.1190 | 0.1120 | 0.1034 |
\(\overline{x}_{12}^{(1)**}\) | 0.1514 | 0.1703 | 0.0423 | 0.2064 | 0.2259 | 0.2100 | 0.2030 | 0.1926 | 0.1890 | 0.1820 | 0.1724 |
\(\overline{x}_{21}^{(1)**}\) | 0.0946 | 0.0690 | 0.0957 | 0.0871 | 0.0947 | 0.1000 | 0.0904 | 0.0770 | 0.0735 | 0.0735 | 0.0690 |
\(\overline{x}_{22}^{(1)**}\) | 0.1081 | 0.1034 | 0.1306 | 0.1419 | 0.1530 | 0.1400 | 0.1238 | 0.1190 | 0.1120 | 0.1120 | 0.1034 |
\(\overline{x}_{31}^{(1)**}\) | 0.1577 | 0.1034 | 0.1913 | 0.1419 | 0.1530 | 0.1600 | 0.1520 | 0.1415 | 0.1190 | 0.1120 | 0.1034 |
\(\overline{x}_{32}^{(1)**}\) | 0.1297 | 0.1379 | 0.2158 | 0.1741 | 0.1984 | 0.1750 | 0.1680 | 0.1582 | 0.1540 | 0.1470 | 0.1379 |
\(\overline{x}_{41}^{(1)**}\) | 0.1892 | 0.1379 | 0.1572 | 0.1741 | 0.1894 | 0.2000 | 0.1920 | 0.1808 | 0.1540 | 0.1470 | 0.1379 |
\(\overline{x}_{42}^{(1)**}\) | 0.1730 | 0.2069 | 0.1590 | 0.2386 | 0.2623 | 0.2450 | 0.2380 | 0.2270 | 0.2240 | 0.2170 | 0.2069 |
\(\overline{x}_{51}^{(1)**}\) | 0.2207 | 0.1724 | 0.1913 | 0.2064 | 0.2259 | 0.2400 | 0.2320 | 0.2201 | 0.1890 | 0.1820 | 0.1724 |
\(\overline{x}_{52}^{(1)**}\) | 0.1730 | 0.2069 | 0.2158 | 0.2386 | 0.2623 | 0.2450 | 0.2380 | 0.2270 | 0.2240 | 0.2170 | 0.2069 |
\(\overline{x}_{11}^{(2)**}\) | 0.1892 | 0.2069 | 0.4438 | 0.1741 | 0.1894 | 0.2000 | 0.1920 | 0.1808 | 0.1540 | 0.1470 | 0.1379 |
\(\overline{x}_{12}^{(2)**}\) | 0.1081 | 0.1202 | 0.0295 | 0.1419 | 0.1530 | 0.1400 | 0.1330 | 0.1238 | 0.1190 | 0.1120 | 0.1034 |
\(\overline{x}_{21}^{(2)**}\) | 0.0946 | 0.0690 | 0.0957 | 0.0871 | 0.0947 | 0.1000 | 0.0960 | 0.0904 | 0.0770 | 0.0735 | 0.690 |
\(\overline{x}_{22}^{(2)**}\) | 0.0432 | 0.0345 | 0.0511 | 0.0548 | 0.1530 | 0.0525 | 0.0490 | 0.0447 | 0.0420 | 0.0385 | 0.0345 |
\(\overline{x}_{31}^{(2)**}\) | 0.2207 | 0.1724 | 0.2255 | 0.2064 | 0.2259 | 0.2400 | 0.2320 | 0.2201 | 0.1890 | 0.1820 | 0.1734 |
\(\overline{x}_{32}^{(2)**}\) | 0.1081 | 0.1034 | 0.1306 | 0.1419 | 0.1530 | 0.1400 | 0.1330 | 0.1238 | 0.1190 | 0.1120 | 0.1034 |
\(\overline{x}_{41}^{(2)**}\) | 0.2207 | 0.1724 | 0.2255 | 0.2064 | 0.2259 | 0.2400 | 0.2320 | 0.2201 | 0.1890 | 0.1820 | 0.1724 |
\(\overline{x}_{42}^{(2)**}\) | 0.1514 | 0.1724 | 0.1874 | 0.2064 | 0.2259 | 0.2100 | 0.2030 | 0.1926 | 0.1890 | 0.1820 | 0.1724 |
\(\overline{x}_{51}^{(2)**}\) | 0.2207 | 0.1724 | 0.2255 | 0.2064 | 0.2259 | 0.2400 | 0.2320 | 0.2201 | 0.1890 | 0.1820 | 0.1724 |
\(\overline{x}_{52}^{(2)**}\) | 0.1297 | 0.1379 | 0.1590 | 0.1741 | 0.1894 | 0.1750 | 0.1680 | 0.1582 | 0.1540 | 0.1470 | 0.1379 |
\(\overline{x}_{11}^{(3)**}\) | 0.1892 | 0.2069 | 0.4438 | 0.1741 | 0.1894 | 0.2000 | 0.1920 | 0.1808 | 0.1540 | 0.1470 | 0.1379 |
\(\overline{x}_{12}^{(3)**}\) | 0.1730 | 0.1953 | 0.0487 | 0.2386 | 0.2623 | 0.2450 | 0.2380 | 0.2270 | 0.2240 | 0.2170 | 0.2069 |
\(\overline{x}_{21}^{(3)**}\) | 0.1577 | 0.1034 | 0.1572 | 0.1419 | 0.1530 | 0.1600 | 0.1520 | 0.1415 | 0.1190 | 0.1120 | 0.1034 |
\(\overline{x}_{22}^{(3)**}\) | 0.1081 | 0.1034 | 0.1306 | 0.1419 | 0.1530 | 0.1400 | 0.1330 | 0.1238 | 0.1190 | 0.1120 | 0.1034 |
\(\overline{x}_{31}^{(3)**}\) | 0.1577 | 0.1034 | 0.1572 | 0.1419 | 0.1530 | 0.1600 | 0.1520 | 0.1415 | 0.1190 | 0.1120 | 0.1034 |
\(\overline{x}_{32}^{(3)**}\) | 0.1946 | 0.2414 | 0.2441 | 0.2709 | 0.2987 | 0.2800 | 0.2730 | 0.2614 | 0.2590 | 0.2520 | 0.2414 |
\(\overline{x}_{41}^{(3)**}\) | 0.2207 | 0.1724 | 0.2255 | 0.2064 | 0.2259 | 0.2400 | 0.3220 | 0.2201 | 0.1890 | 0.1820 | 0.1724 |
\(\overline{x}_{42}^{(3)**}\) | 0.1081 | 0.1034 | 0.1306 | 0.1419 | 0.1530 | 0.1400 | 0.1330 | 0.1238 | 0.1190 | 0.1120 | 0.1034 |
\(\overline{x}_{51}^{(3)**}\) | 0.1892 | 0.1379 | 0.1913 | 0.1741 | 0.1894 | 0.2000 | 0.1920 | 0.1808 | 0.1540 | 0.1470 | 0.1379 |
\(\overline{x}_{52}^{(3)**}\) | 0.1514 | 0.1724 | 0.1874 | 0.2064 | 0.2259 | 0.2100 | 0.2030 | 0.1926 | 0.1890 | 0.1820 | 0.1724 |
OSO | \(\alpha\) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
0.0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | |
\(\overline{y}_{11}^{(1)**}\) | 0.1568 | 0.1721 | 0.0423 | 0.1991 | 0.2168 | 0.2250 | 0.2080 | 0.2075 | 0.1800 | 0.1710 | 0.1576 |
\(\overline{y}_{21}^{(1)**}\) | 0.0000 | 0.0156 | 0.0070 | 0.0459 | 0.0619 | 0.0750 | 0.0780 | 0.0854 | 0.0800 | 0.0810 | 0.0788 |
\(\overline{y}_{31}^{(1)**}\) | 0.0000 | 0.0156 | 0.0070 | 0.0459 | 0.0619 | 0.0750 | 0.0780 | 0.0854 | 0.0800 | 0.0810 | 0.0788 |
\(\overline{y}_{41}^{(1)**}\) | 0.0784 | 0.0939 | 0.0247 | 0.1225 | 0.1393 | 0.1500 | 0.1430 | 0.1465 | 0.1300 | 0.1260 | 0.0690 |
\(\overline{y}_{51}^{(1)**}\) | 0.1568 | 0.1721 | 0.0423 | 0.1991 | 0.2168 | 0.2250 | 0.2080 | 0.2075 | 0.1800 | 0.1710 | 0.1576 |
\(\overline{y}_{11}^{(2)**}\) | 0.0865 | 0.1014 | 0.0269 | 0.1354 | 0.1475 | 0.1600 | 0.1540 | 0.1587 | 0.1300 | 0.1260 | 0.1034 |
\(\overline{y}_{21}^{(2)**}\) | 0.0000 | 0.0169 | 0.0077 | 0.0508 | 0.0656 | 0.0800 | 0.0840 | 0.0926 | 0.0800 | 0.0810 | 0.0690 |
\(\overline{y}_{31}^{(2)**}\) | 0.0000 | 0.0169 | 0.0477 | 0.0508 | 0.0656 | 0.0800 | 0.0840 | 0.0926 | 0.0800 | 0.0810 | 0.0690 |
\(\overline{y}_{41}^{(2)**}\) | 0.1730 | 0.0185 | 0.0461 | 0.2201 | 0.2295 | 0.2400 | 0.2240 | 0.2248 | 0.1800 | 0.1710 | 0.1379 |
\(\overline{y}_{51}^{(2)**}\) | 0.1730 | 0.1859 | 0.0461 | 0.2201 | 0.2295 | 0.2400 | 0.2240 | 0.2248 | 0.1800 | 0.1710 | 0.1379 |
\(\overline{y}_{11}^{(1)(O)**}\) | 0.0784 | 0.0861 | 0.0211 | 0.0996 | 0.1084 | 0.1125 | 0.1040 | 0.1038 | 0.0900 | 0.0855 | 0.0788 |
\(\overline{y}_{21}^{(1)(O)**}\) | 0.0000 | 0.0078 | 0.0038 | 0.0230 | 0.0310 | 0.0375 | 0.0390 | 00,427 | 0.0400 | 0.0405 | 0.0394 |
\(\overline{y}_{31}^{(1)(O)**}\) | 0.0000 | 0.0085 | 0.0038 | 0.0254 | 00,328 | 0.0400 | 0.0420 | 0.0463 | 0.0400 | 0.0405 | 0.0345 |
\(\overline{y}_{41}^{(1)(O)**}\) | 0.0784 | 0.0861 | 0.0211 | 0.0996 | 0.1084 | 0.1125 | 0.1040 | 0.1038 | 0.0900 | 0.0855 | 0.0788 |
\(\overline{y}_{51}^{(1)(O)**}\) | 0.1568 | 0.1643 | 0.0387 | 0.1761 | 0.1858 | 0.1875 | 0.1690 | 0.1648 | 0.1400 | 0.1305 | 0.1182 |
\(\overline{y}_{11}^{(1)(I)**}\) | 0.2351 | 0.2269 | 0.0493 | 0.2068 | 0.2013 | 0.1875 | 0.1560 | 0.1404 | 0.1100 | 0.0945 | 0.0788 |
\(\overline{y}_{21}^{(1)(I)**}\) | 0.1568 | 0.1487 | 0.0317 | 0.1302 | 0.1239 | 0.1125 | 0.0910 | 0.0793 | 0.0600 | 0.0495 | 0.0394 |
\(\overline{y}_{31}^{(1)(I)**}\) | 0.1568 | 0.1487 | 0.0317 | 0.1302 | 0.1239 | 0.1125 | 0.0910 | 0.0193 | 0.0600 | 0.0495 | 0.0394 |
\(\overline{y}_{41}^{(1)(I)**}\) | 0.1568 | 0.1487 | 0.0170 | 0.1302 | 0.1239 | 0.1125 | 0.0910 | 0.0793 | 0.0600 | 0.0495 | 0.0394 |
\(\overline{y}_{51}^{(1)(I)**}\) | 0.1568 | 0.1487 | 0.0317 | 0.1302 | 0.1239 | 0.1125 | 0.0910 | 0.0793 | 0.0600 | 0.0495 | 0.0394 |
\(\overline{y}_{11}^{(2)(O)**}\) | 0.0865 | 0.0930 | 0.0231 | 0.1100 | 0.1148 | 0.1200 | 0.1120 | 0.1124 | 0.0900 | 0.0855 | 0.0690 |
\(\overline{y}_{21}^{(2)(O)**}\) | 0.0000 | 0.0085 | 0.0038 | 0.0254 | 0.0328 | 0.0400 | 0.0420 | 0.0463 | 0.0400 | 0.0405 | 0.0345 |
\(\overline{y}_{31}^{(2)(O)**}\) | 0.0000 | 0.0085 | 0.0038 | 0.0254 | 0.0328 | 0.0400 | 0.0420 | 0.0463 | 0.0400 | 0.0405 | 0.0345 |
\(\overline{y}_{41}^{(2)(O)**}\) | 0.1730 | 0.1775 | 0.0423 | 0.1947 | 0.1967 | 0.2000 | 0.1820 | 0.1785 | 0.1400 | 0.1305 | 0.1034 |
\(\overline{y}_{51}^{(2)(O)**}\) | 0.0865 | 0.0930 | 0.0231 | 0.1100 | 0.1148 | 0.1200 | 0.1120 | 0.1124 | 0.0900 | 0.0855 | 0.0690 |
\(\overline{y}_{11}^{(2)(I)**}\) | 0.1730 | 0.1606 | 0.0346 | 0.1439 | 0.1311 | 0.1200 | 0.0980 | 0.0860 | 0.0600 | 0.0495 | 0.0345 |
\(\overline{y}_{21}^{(2)(I)**}\) | 0.1730 | 0.1606 | 0.0346 | 0.1439 | 0.1311 | 0.1200 | 0.0980 | 0.0860 | 0.0600 | 0.0495 | 0.0345 |
\(\overline{y}_{31}^{(2)(I)**}\) | 0.1730 | 0.1606 | 0.0346 | 0.0439 | 0.1311 | 0.1200 | 0.0980 | 0.0860 | 0.0600 | 0.0495 | 0.0345 |
\(\overline{y}_{41}^{(2)(I)**}\) | 0.1730 | 0.1606 | 0.0346 | 0.1439 | 0.1311 | 0.1200 | 0.0980 | 0.0860 | 0.0600 | 0.0495 | 0.0345 |
\(\overline{y}_{51}^{(2)(I)**}\) | 0.2595 | 0.2451 | 0.0538 | 0.2285 | 0.2131 | 0.2000 | 0.1680 | 0.1521 | 0.1100 | 0.0945 | 0.0690 |
\(\overline{y}_{11}^{(3)**}\) | 0.0000 | 0.0354 | 0.0149 | 0.0931 | 0.1166 | 0.1325 | 0.1290 | 0.1238 | 0.1200 | 0.1125 | 0.1116 |
\(\overline{y}_{21}^{(3)**}\) | 0.0000 | 0.0354 | 0.0149 | 0.0931 | 0.1166 | 0.1325 | 0.1290 | 0.1238 | 0.1200 | 0.1125 | 0.1116 |
\(\overline{y}_{31}^{(3)**}\) | 0.3784 | 0.3890 | 0.0893 | 0.04035 | 0.04080 | 0.3975 | 0.3440 | 0.3007 | 0.2700 | 0.2375 | 0.2233 |
\(\overline{y}_{41}^{(3)**}\) | 0.0000 | 0.0354 | 0.0149 | 0.0931 | 0.1166 | 0.1325 | 0.1290 | 0.1238 | 0.1200 | 0.1125 | 0.1116 |
\(\overline{y}_{51}^{(3)**}\) | 0.7568 | 0.7427 | 0.1638 | 0.7138 | 0.6995 | 0.6625 | 0.5590 | 0.4776 | 0.4200 | 0.03625 | 0.3349 |
DMU | \(\alpha\) | 0.0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
A | P1 | 0.9090 | 0.9105 | 0.9168 | 0.9282 | 0.9389 | 0.9480 | 0.9550 | 0.9742 | 0.9880 | 0.9940 | 1 |
P2 | 0.5273 | 0.5371 | 0.5633 | 0.5860 | 0.5924 | 0.6100 | 0.6250 | 0.6347 | 0.6430 | 0.6590 | 0.6412 | |
P3 | 0.2003 | 0.2197 | 0.2264 | 0.2334 | 0.2542 | 0.2625 | 0.2740 | 0.2792 | 0.2880 | 0.2965 | 0.2989 | |
S1 | 0.7295 | 0.7390 | 0.7458 | 0.7507 | 0.7796 | 0.8075 | 0.8105 | 0.8277 | 0.8330 | 0.8540 | 0.8621 | |
S2 | 0.4054 | 0.4102 | 0.4144 | 0.4273 | 0.4460 | 0.4502 | 0.4620 | 0.4741 | 0.4880 | 0.4975 | 0.5059 | |
S | 0.2957 | 0.3031 | 0.3091 | 0.3208 | 0.3477 | 0.3635 | 0.3744 | 0.3924 | 0.4065 | 0.4249 | 0.4361 | |
B | P1 | 0.6327 | 0.6436 | 0.6514 | 0.6689 | 0.6877 | 0.7085 | 0.7290 | 0.7543 | 0.7760 | 0.7855 | 0.7924 |
P2 | 0.8278 | 0.8310 | 0.8534 | 0.8719 | 0.8800 | 0.9026 | 0.9250 | 0.9651 | 0.9810 | 0.9920 | 1 | |
P3 | 0.3455 | 0.3506 | 0.3603 | 0.3778 | 0.3810 | 0.3925 | 0.4240 | 0.4370 | 0.4580 | 0.4685 | 0.4725 | |
S1 | 0.6126 | 0.6611 | 0.6940 | 0.7160 | 0.7789 | 0.8125 | 0.8350 | 0.8396 | 0.8480 | 0.8690 | 0.6828 | |
S2 | 0.3258 | 0.3340 | 0.3533 | 0.3805 | 0.4335 | 0.4550 | 0.4652 | 0.4734 | 0.4980 | 0.5060 | 0.7547 | |
S | 0.1995 | 0.2208 | 0.2451 | 0.2724 | 0.3377 | 0.3697 | 0.3884 | 0.3975 | 0.4223 | 0.4397 | 0.4543 | |
C | P1 | 0.4806 | 0.4846 | 0.4904 | 0.4960 | 0.5024 | 0.5189 | 0.5200 | 0.5398 | 0.5530 | 0.5614 | 0.5684 |
P2 | 0.2916 | 0.2975 | 0.3025 | 0.3082 | 0.3178 | 0.3289 | 0.3350 | 0.3440 | 0.3580 | 0.3685 | 0.3742 | |
P3 | 0.5102 | 0.5264 | 0.5315 | 0.5386 | 0.5467 | 0.5525 | 0.5540 | 0.5582 | 0.6680 | 0.6805 | 0.6875 | |
S1 | 0.6081 | 0.6102 | 0.6178 | 0.6227 | 0.6296 | 0.6348 | 0.6400 | 0.6477 | 0.6530 | 0.6723 | 0.6789 | |
S2 | 0.6907 | 0.6981 | 0.7044 | 0.7109 | 0.7192 | 0.7215 | 0.7279 | 0.7310 | 0.7380 | 0.7460 | 0.7538 | |
S | 0.4200 | 0.4260 | 0.4352 | 0.4227 | 0.4581 | 0.4580 | 0.4659 | 0.4735 | 0.4819 | 0.5015 | 0.5118 | |
D | P1 | 0.4625 | 0.4812 | 0.4874 | 0.4923 | 0.4985 | 0.5150 | 0.5276 | 0.5302 | 0.5562 | 0.5736 | 0.5985 |
P2 | 0.4701 | 0.4745 | 0.4814 | 0.4874 | 0.4915 | 0.4991 | 0.5146 | 0.5214 | 0.5324 | 0.5514 | 0.6015 | |
P3 | 0.2914 | 0.2985 | 0.3025 | 0.3225 | 0.3365 | 0.3412 | 0.3519 | 0.3581 | 0.3626 | 0.3713 | 0.4012 | |
S1 | 0.5726 | 0.5814 | 0.5917 | 0.6104 | 0.6478 | 0.6521 | 0.6655 | 0.6700 | 0.6712 | 0.6812 | 0.7139 | |
S2 | 0.4843 | 0.4912 | 0.5024 | 0.5217 | 0.5485 | 0.5512 | 0.5813 | 0.5972 | 0.5985 | 0.6025 | 0.6525 | |
S | 0.2773 | 0.2856 | 0.2973 | 0.3184 | 0.3553 | 0.3594 | 0.3866 | 0.4001 | 0.4017 | 0.4104 | 0.4658 | |
E | P1 | 0.5369 | 0.5549 | 0.5717 | 0.5829 | 0.5917 | 0.6022 | 0.6107 | 0.6218 | 0.6512 | 0.7015 | 0.7213 |
P2 | 0.4985 | 0.5128 | 0.5289 | 0.5312 | 0.5512 | 0.5816 | 0.5874 | 0.6025 | 0.6215 | 0.6412 | 0.6512 | |
P3 | 0.9025 | 0.9205 | 0.9316 | 0.9415 | 0.9465 | 0.9652 | 0.9714 | 0.9784 | 0.9857 | 1 | 1 | |
S1 | 0.6625 | 0.6874 | 0.6919 | 0.7098 | 0.7145 | 0.7284 | 0.7305 | 0.7355 | 0.7418 | 0.7512 | 0.7836 | |
S2 | 0.9217 | 0.9325 | 0.9415 | 0.9474 | 0.9602 | 0.9725 | 0.9813 | 0.9863 | 0.9924 | 1 | 1 | |
S | 0.6106 | 0.6410 | 0.6514 | 0.6725 | 0.6861 | 0.7084 | 0.7168 | 0.7254 | 0.7362 | 0.7512 | 0.7836 |
Sensitive and comparison analysis
Sensitivity of efficiency results to alpha
Impact of fuzzy data on ranking and efficiency results
\(\alpha = 0\) | P1 | A ≻ B ≻ E ≻ C ≻ D | \(\alpha = 0.1\) | P1 | A≻ B ≻ E ≻C ≻D |
P2 | B ≻ A ≻ E ≻ D ≻ C | P2 | B ≻A ≻ E ≻ D ≻ C | ||
P3 | E ≻ C ≻ B ≻ D ≻ A | P3 | E ≻C≻ B ≻ D ≻ A | ||
S1 | A ≻ E ≻ B ≻ C ≻ D | S1 | A≻ E ≻ B≻ C ≻ D | ||
S2 | E ≻ C ≻ D ≻ A ≻ B | S2 | E ≻ C≻ D ≻ A ≻ B | ||
S | E ≻ C ≻ A ≻ D ≻ B | S | E ≻C ≻ A ≻ D ≻B | ||
\(\alpha = 0.2\) | P1 | A ≻ B ≻ E ≻ C ≻ D | \(\alpha = 0.3\) | P1 | A≻ B ≻ E ≻C ≻D |
P2 | B ≻ A ≻ E ≻ D ≻ C | P2 | B ≻A ≻ E ≻ D ≻ C | ||
P3 | E ≻ C ≻ B ≻ D ≻ A | P3 | E ≻C≻ B ≻ D ≻ A | ||
S1 | A ≻ B ≻ E ≻ C ≻ D | S1 | A≻ B ≻ E≻ C ≻ D | ||
S2 | E ≻ C ≻ D ≻ A ≻ B | S2 | E ≻ C≻ D ≻ A ≻ B | ||
S | E ≻ C ≻ A ≻ D ≻ B | S | E ≻C ≻ A ≻ D ≻B | ||
\(\alpha = 0.4\) | P1 | A ≻ B ≻ E ≻ C ≻ D | \(\alpha = 0.5\) | P1 | A≻ B ≻ E ≻C ≻D |
P2 | B ≻ A ≻ E ≻ D ≻ C | P2 | B ≻A ≻ E ≻ D ≻ C | ||
P3 | E ≻ C ≻ B ≻ D ≻ A | P3 | E ≻C≻ B ≻ D ≻ A | ||
S1 | A ≻ B ≻ E ≻ D ≻ C | S1 | A≻ B ≻ E≻ D≻ C | ||
S2 | E ≻ C ≻ D ≻ A ≻ B | S2 | E ≻ C≻ D ≻ B ≻ A | ||
S | E ≻ C ≻ D ≻ A ≻ B | S | E ≻ C≻B≻A≻ D | ||
\(\alpha = 0.6\) | P1 | A ≻ B ≻ E ≻ D ≻ C | \(\alpha = 0.7\) | P1 | A≻ B ≻ E ≻C ≻D |
P2 | B ≻ A ≻ E ≻ D ≻ C | P2 | B ≻A ≻ E ≻ D ≻ C | ||
P3 | E ≻ C ≻ B ≻ D ≻ A | P3 | E ≻C≻ B ≻ D ≻ A | ||
S1 | B ≻ A ≻ E ≻ D ≻ C | S1 | B≻ A ≻ E≻ D ≻ C | ||
S2 | E ≻ C ≻ D ≻ B ≻ A | S2 | E ≻ C≻ D ≻ A ≻ B | ||
S | E ≻ C ≻ B ≻ D ≻ A | S | E ≻C ≻ D ≻ B ≻ A | ||
\(\alpha = 0.8\) | P1 | A ≻ B ≻ E ≻ D ≻ C | \(\alpha = 0.9\) | P1 | A≻ B ≻ E ≻D≻C |
P2 | B ≻ A ≻ E ≻ D ≻ C | P2 | B ≻A ≻ E ≻ D ≻ C | ||
P3 | E ≻ C ≻ B ≻ D ≻ A | P3 | E ≻C≻ B ≻ D ≻ A | ||
S1 | B ≻ A ≻ E ≻ D ≻ C | S1 | B≻ A ≻ E≻ D ≻ C | ||
S2 | E ≻ C ≻ D ≻ B ≻ A | S2 | E ≻ C≻ D ≻ B ≻ A | ||
S | E ≻ C ≻ B ≻ A ≻ D | S | E ≻C ≻ B ≻ A ≻D | ||
\(\alpha = 1\) | P1 | A ≻ B ≻ E ≻ D ≻ C | |||
P2 | B ≻ A ≻ E ≻ D ≻ C | ||||
P3 | E ≻ C ≻ B ≻ D ≻ A | ||||
S1 | A ≻ E ≻ D ≻ C ≻ B | ||||
S2 | E ≻ B ≻ C ≻ D ≻ A | ||||
S | E ≻ B ≻ C ≻ D ≻ A |
Comparison between results of fuzzy CCR model and fuzzy relational NDEA model
Computational and technical comparisons
DMU | System | Process 1 | Process 2 | Process 3 | Stage 1 | Stage 2 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
K | HA | K | HA | K | HA | K | HA | K | HA | K | HA | |
A | 0.4744 | 0.4667 | 1 | 1 | 0.6613 | 0.6429 | 0.3070 | 0.3011 | 0.9013 | 0.9000 | 0.5264 | 0.5185 |
B | 0.5895 | 0.5833 | 0.8188 | 0.8000 | 1 | 1 | 0.5003 | 0.4912 | 0.9286 | 0.9286 | 0.6348 | 0.6282 |
C | 0.5209 | 0.5133 | 0.5618 | 0.5714 | 0.3796 | 0.3750 | 0.7204 | 0.6914 | 0.6677 | 0.6800 | 0.7801 | 0.7549 |
D | 0.4706 | 0.7402 | 0.5990 | 0.6000 | 0.6069 | 0.6000 | 0.4029 | 0.4058 | 0.7174 | 0.7143 | 0.6561 | 0.6583 |
E | 0.7931 | 0.7931 | 0.7273 | 0.7273 | 0.667 | 0.6667 | 1 | 1 | 0.7931 | 0.7931 | 1 | 1 |
Application in oil and gas refineries
DMU | \({X}_{1}\) | \({X}_{2}\) | \({Y}_{1}^{(O)}\) | \({Y}_{1}^{(I)}\) | \({Y}_{2}^{(O)}\) | \({Y}_{2}^{(I)}\) | \({Y}_{3}\) | |
---|---|---|---|---|---|---|---|---|
Refinery1 | System | (96, 98, 100) | (85, 86, 87) | – | – | |||
Process 1 | (33.6, 34.3, 35) | (29.7, 30, 30.4) | (46, 47, 48) | (11, 11.2, 11.4) | – | – | – | |
Process 2 | (38.4, 39, 40) | (34, 34.4, 34.8) | – | (85, 93, 106) | (20, 22, 25) | – | ||
Process 3 | (24, 24.5, 25) | (21, 21.5, 21.7) | – | – | (50, 53, 54) | |||
Refinery2 | System | (75, 78, 81) | (86, 88, 90) | – | – | |||
Process 1 | (26.2, 27.3, 28.3) | (30, 31.8, 31.5) | (29, 30, 32) | (7, 7.2, 7.6) | – | – | – | |
Process 2 | (30, 31.2, 32) | (34.4, 35.2, 36) | – | (68, 74, 85) | (16, 17, 20) | – | ||
Process 3 | (18.7, 19.5, 20) | (21.5, 22, 22.5) | – | – | (51, 54, 59) | |||
Refinery3 | System | (77, 78, 80) | (85, 87, 89) | – | – | |||
Process 1 | (26.9, 27.3, 28) | (29.7, 30, 31) | (30, 32, 32.3) | (7.2, 7, 7.6) | – | – | – | |
Process 2 | (30.8, 31.2, 32) | (34, 34.8, 35) | – | – | (68, 76, 85) | (16, 18, 20) | – | |
Process 3 | (19.2, 19.5, 20) | (21.2, 21.7, 22) | – | – | (50, 54, 56) | |||
Refinery4 | System | (91, 92, 94) | (93, 94, 96) | – | – | |||
Process 1 | (31.8, 32.2, 32.9) | (32.5, 33, 33.6) | (28, 29, 31) | (6.6, 7, 7.4) | – | – | – | |
Process 2 | (36.4, 36.8, 37) | (37, 37.6, 38) | – | (79, 86, 91) | (18, 20, 21) | – | ||
Process 3 | (22.7, 23, 23.5) | (23.2, 23.5, 24) | – | – | (54, 57, 58) | |||
Refinery5 | System | (89, 90, 92) | (82, 83, 84) | – | – | |||
Process 1 | (31, 31.5, 32.2) | (28.7, 29, 29.4) | (44, 45, 47) | (10, 10.6, 11) | – | – | – | |
Process 2 | (35.6, 36, 36.8) | (32.8, 33, 33.6) | – | (78, 83, 92) | (18, 19, 21) | – | ||
Process 3 | (22, 22.5, 23) | (20, 20.75, 21) | – | – | (52, 59, 59.4) | |||
Refinery6 | System | (102, 103, 105) | (96, 97, 99) | |||||
Process1 | (35.7, 36, 36.7) | (33.6, 34, 34.6) | (59, 61, 63) | (14, 14.4, 15) | ||||
Process2 | (35.6, 36, 36.8) | (38, 38.8, 39.6) | (89, 93, 96) | (21, 22, 22.8) | ||||
Process3 | (25, 25.75, 26.2) | (24, 24.2, 24.7) | – | (55, 59, 61) | ||||
Refinery7 | System | (96, 97, 100) | (90, 91, 92) | |||||
Process 1 | (33.6, 33.9, 35) | (31, 31.8, 32) | (25, 29, 34) | (6.4, 7, 8.2) | ||||
Process 2 | (40.8, 41, 42) | (36, 36.4, 36.8)) | (85, 92, 102) | (20, 21.8, 24) | ||||
Process 3 | (24, 24.25, 25) | (22.5, 22.7, 23) | (50, 52, 55) | |||||
Refinery8 | System | (85, 87, 90) | (90, 92, 93) | |||||
Process 1 | (29.7, 30, 31.5) | (31.5, 32, 32.5) | (33, 38, 39) | (8, 9, 9.2) | ||||
Process 2 | (38.4, 38.8, 40) | (36, 36.8, 37) | (76, 79, 83) | (18, 18, 8, 19) | ||||
Process 3 | (21, 21.75, 22.5) | (22, 23, 23.2) | (90, 96, 97) | |||||
Refinery9 | System | (106, 108, 112) | (84, 88, 92) | |||||
Process 1 | (37, 37.8, 39.2) | (29, 30.8, 32) | (68, 73, 74) | (17, 17.2, 17.6) | ||||
Process 2 | (34, 34.8, 36) | (33.6, 35, 36.8) | (95, 100, 107) | (22, 23, 25) | ||||
Process 3 | (26.5, 27, 28) | (21, 22, 23) | (52, 55, 57) | |||||
Refinery10 | System | (107, 108, 111) | (90, 95, 97) | |||||
Process 1 | (37.4, 37.8, 38.8) | (31.5, 33, 33.9) | (36, 39, 40) | (9.2, 9, 9.4) | ||||
Process 2 | (42.4, 43, 44.8) | (36, 38, 38.8)) | (94, 96, 99) | (22, 22.6, 23) | ||||
Process 3 | (26.7, 27, 27.7) | (22.5, 23.7, 24) | (51, 54, 58) | |||||
Refinery11 | System | (94, 97, 101) | (76, 78, 79) | |||||
Process 1 | (32.9, 33.9, 35.3) | (26, 27.3, 27.6) | (28, 34, 37) | (7, 8, 8.8) | ||||
Process 2 | (37.6, 38.8, 40) | (30, 31.2, 31.6) | (85, 94, 100) | (20, 22, 23) | ||||
Process 3 | (23.5, 24.2, 25, 2) | (19, 19.5, 19.7) | (55, 57, 59) | |||||
Refinery12 | System | (77, 78, 79) | (86, 89, 90) | |||||
Process 1 | (26.9, 27.3, 27.6) | (30, 31, 31.5) | (31, 33, 34) | (7, 7.8, 8.2) | ||||
Process 2 | (30, 31.2, 31.6) | (34, 35.6, 36) | (67, 74, 78) | (15, 17, 18) | ||||
Process 3 | (19.2, 19.5, 19.7) | (21.5, 22, 22.5) | (59, 60, 61) | |||||
Refinery13 | System | (100, 102, 105) | (105, 107, 111) | |||||
Process 1 | (35, 35.7, 36.7) | (36.7, 37, 38.8) | (44, 48, 50) | (11, 11.4, 11.8) | ||||
Process 2 | (40, 40.8, 42) | (42, 42.8, 44.4) | (86, 94, 97) | (20, 22, 23) | ||||
Process 3 | (25, 25.5, 26.2) | (26, 26.7, 27.7) | (102, 107, 109) | |||||
Refinery14 | System | (82, 85, 88) | (92, 93, 94) | |||||
Process 1 | (28, 29.7, 30.8) | (36.8, 37, 37.6) | (69, 74, 75) | (17, 17.6, 17.8) | ||||
Process 2 | (32.8, 34, 35.2) | (32.8, 34, 35.2) | (74, 85, 90) | (17, 20, 21) | ||||
Process 3 | (20.5, 21.2, 22) | (23, 23.2, 23.5) | (78, 79, 82) | |||||
Refinery15 | System | (77, 80, 82) | (90, 92, 94) | |||||
Process 1 | (26.9, 28, 28.7) | (31.5, 32, 32.9) | (40, 42, 45) | (10, 10, 10.6) | ||||
Process 2 | (30.8, 32, 32.8) | (36, 36.8, 37) | (69, 74, 80) | (16, 17, 19) | ||||
Process 3 | (19.2, 20, 20.5) | (22.5, 23, 23.5) | (62, 65, 67) |
\(\alpha\) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
CWS | 0.0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 |
\(v_{1}^{**}\) | 0.046 | 0.046 | 0.046 | 0.047 | 0.047 | 0.047 | 0.046 | 0.047 | 0.047 | 0.047 | 0.046 |
\(v_{2}^{**}\) | 0.048 | 0.048 | 0.048 | 0.048 | 0.048 | 0.047 | 0.048 | 0.047 | 0.047 | 0.047 | 0.046 |
\(u_{1}^{(1)**}\) | 0.018 | 0.018 | 0.018 | 0.018 | 0.018 | 0.018 | 0.018 | 0.018 | 0.018 | 0.018 | 0.017 |
\(u_{1}^{(2)**}\) | 0.019 | 0.019 | 0.019 | 0.018 | 0.018 | 0.018 | 0.019 | 0.017 | 0.017 | 0.017 | 0.016 |
\(u_{1}^{(3)**}\) | 0.018 | 0.018 | 0.018 | 0.018 | 0.018 | 0.017 | 0.018 | 0.017 | 0.017 | 0.017 | 0.016 |
E | \(\alpha\) | 0.0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Refinery 1 | P1 | 0.382 | 0.372 | 0.376 | 0.380 | 0.385 | 0.388 | 0.390 | 0.395 | 0.397 | 0.392 | 0.402 |
P2 | 0.760 | 0.756 | 0.780 | 0.782 | 0.785 | 0.788 | 0.790 | 0.792 | 0.810 | 0.812 | 0.825 | |
P3 | 0.660 | 0.665 | 0.670 | 0.672 | 0.675 | 0.680 | 0.681 | 0.684 | 0.667 | 0.675 | 0.690 | |
S | 0.363 | 0.3605 | 0.3517 | 0.3769 | 0.3771 | 0.3848 | 0.3851 | 0.3801 | 0.3891 | 0.3956 | 0.4023 | |
Refinery 2 | P1 | 0.370 | 0.365 | 0.362 | 0.372 | 0.374 | 0.376 | 0.380 | 0.382 | 0.384 | 0.383 | 0.392 |
P2 | 0.745 | 0.748 | 0.743 | 0.752 | 0.756 | 0.758 | 0.751 | 0.762 | 0.765 | 0.763 | 0.771 | |
P3 | 0.650 | 0.645 | 0.652 | 0.658 | 0.654 | 0.662 | 0.664 | 0.639 | 0.648 | 0.672 | 0.674 | |
S | 0.362 | 0.375 | 0.350 | 0.362 | 0.356 | 0.369 | 0.375 | 0.376 | 0.379 | 0.378 | 0.388 | |
Refinery 3 | P1 | 0.353 | 0.354 | 0.356 | 0.358 | 0.361 | 0.367 | 0.373 | 0.375 | 0.368 | 0.374 | 0.380 |
P2 | 0.732 | 0.736 | 0.731 | 0.745 | 0.742 | 0.751 | 0.753 | 0.766 | 0.764 | 0.748 | 0.785 | |
P3 | 0.621 | 0.625 | 0.632 | 0.631 | 0.635 | 0.641 | 0.647 | 0.645 | 0.650 | 0.648 | 0.652 | |
S | 0.3411 | 0.3457 | 0.3382 | 0.3381 | 0.3416 | 0.3429 | 0.3425 | 0.355 | 0.3503 | 0.3476 | 0.3597 | |
Refinery 4 | P1 | 0.410 | 0.415 | 0.425 | 0.430 | 0.418 | 0.422 | 0.432 | 0.436 | 0.428 | 0.440 | 0.448 |
P2 | 0.780 | 0.782 | 0.786 | 0.792 | 0.805 | 0.808 | 0.810 | 0.795 | 0.820 | 0.815 | 0.820 | |
P3 | 0.652 | 0.656 | 0.632 | 0.644 | 0.658 | 0.662 | 0.663 | 0.654 | 0.668 | 0.670 | 0.672 | |
S | 0.391 | 0.3945 | 0.3834 | 0.3845 | 0.3796 | 0.3849 | 0.3854 | 0.3852 | 0.3897 | 0.399 | 0.4071 | |
Refinery 5 | P1 | 0.480 | 0.482 | 0.490 | 0.487 | 0.492 | 0.495 | 0.483 | 0.496 | 0.505 | 0.510 | 0.512 |
P2 | 0.735 | 0.740 | 0.733 | 0.738 | 0.742 | 0.746 | 0.748 | 0.750 | 0.742 | 0.740 | 0.755 | |
P3 | 0.651 | 0.648 | 0.655 | 0.653 | 0.660 | 0.658 | 0.662 | 0.665 | 0.654 | 0.660 | 0.663 | |
S | 0.4220 | 0.4275 | 0.4236 | 0.4131 | 0.4106 | 0.4267 | 0.4136 | 0.4309 | 0.4243 | 0.4451 | 0.4518 | |
Refinery 6 | P1 | 0.515 | 0.522 | 0.525 | 0.518 | 0.520 | 0.532 | 0.527 | 0.512 | 0.523 | 0.526 | 0.530 |
P2 | 0.754 | 0.748 | 0.758 | 0.762 | 0.763 | 0.756 | 0.758 | 0.760 | 0.757 | 0.762 | 0.770 | |
P3 | 0.662 | 0.665 | 0.656 | 0.653 | 0.648 | 0.646 | 0.652 | 0.663 | 0.670 | 0.661 | 0.663 | |
S | 0.391 | 0.3945 | 0.3834 | 0.3845 | 0.3796 | 0.3849 | 0.3854 | 0.3852 | 0.38976 | 0.399 | 0.4071 | |
Refinery 7 | P1 | 0.333 | 0.324 | 0.325 | 0.328 | 0.341 | 0.343 | 0.348 | 0.352 | 0.356 | 0.364 | 0.372 |
P2 | 0.702 | 0.716 | 0.721 | 0.725 | 0.732 | 0.745 | 0.748 | 0.752 | 0.755 | 0.738 | 0.768 | |
P3 | 0.601 | 0.612 | 0.622 | 0.620 | 0.615 | 0.618 | 0.632 | 0.627 | 0.633 | 0.641 | 0.645 | |
S | 0.3317 | 0.3369 | 0.3339 | 0.3403 | 0.3213 | 0.3251 | 0.3296 | 0.3340 | 0.3354 | 0.3452 | 0.3445 | |
Refinery 8 | P1 | 0.585 | 0.587 | 0.590 | 0.580 | 0.592 | 0.583 | 0.586 | 0.594 | 0.593 | 0.580 | 0.598 |
P2 | 0.810 | 0.816 | 0.824 | 0.821 | 0.832 | 0.834 | 0.826 | 0.837 | 0.817 | 0.824 | 0.835 | |
P3 | 0.701 | 0.721 | 0.718 | 0.732 | 0.740 | 0.732 | 0.734 | 0.729 | 0.722 | 0.731 | 0.729 | |
S | 0.4410 | 0.4461 | 0.4458 | 0.4442 | 0.4426 | 0.4464 | 0.4555 | 0.4418 | 0.4406 | 0.4438 | 0.4599 | |
Refinery 9 | P1 | 0.605 | 0.605 | 0.612 | 0.617 | 0.610 | 0.608 | 0.615 | 0.618 | 0.614 | 0.618 | 0.620 |
P2 | 0.845 | 0.855 | 0.867 | 0.835 | 0.875 | 0.882 | 0.867 | 0.870 | 0.892 | 0.895 | 0.898 | |
P3 | 0.751 | 0.720 | 0.745 | 0.746 | 0.740 | 0.765 | 0.742 | 0.780 | 0.785 | 0.790 | 0.795 | |
S | 0.4611 | 0.4692 | 0.4750 | 0.2535 | 0.4636 | 0.4706 | 0.4746 | 0.4717 | 0.4547 | 0.4500 | 0.4847 | |
Refinery 10 | P1 | 0.430 | 0.435 | 0.438 | 0.440 | 0.432 | 0.436 | 0.442 | 0.445 | 0.447 | 0.450 | 0.453 |
P2 | 0.812 | 0.815 | 0.817 | 0.823 | 0.827 | 0.832 | 0.836 | 0.739 | 0.842 | 0.844 | 0.845 | |
P3 | 0.652 | 0.656 | 0.632 | 0.644 | 0.658 | 0.662 | 0.663 | 0.654 | 0.668 | 0.670 | 0.672 | |
S | 0.402 | 0.400 | 0.403 | 0.411 | 0.414 | 0.418 | 0.418 | 0.420 | 0.421 | 0.426 | 0.436 | |
Refinery 11 | P1 | 0.470 | 0.472 | 0.480 | 0.467 | 0.462 | 0.465 | 0.475 | 0.476 | 0.480 | 0.482 | 0.485 |
P2 | 0.715 | 0.720 | 0.723 | 0.728 | 0.732 | 0.726 | 0.725 | 0.730 | 0.735 | 0.715 | 0.730 | |
P3 | 0.610 | 0.618 | 0.625 | 0.632 | 0.641 | 0.636 | 0.645 | 0.620 | 0.628 | 0.636 | 0.640 | |
S | 0.419 | 0.4281 | 0.4283 | 0.435 | 0.4361 | 0.4334 | 0.4342 | 0.4342 | 0.4431 | 0.4428 | 0.4482 | |
Refinery 12 | P1 | 0.452 | 0.453 | 0.457 | 0.456 | 0.460 | 0.462 | 0.450 | 0.462 | 0.465 | 0.457 | 0.467 |
P2 | 0.701 | 0.706 | 0.708 | 0.711 | 0.715 | 0.714 | 0.718 | 0.703 | 0.704 | 0.712 | 0.710 | |
P3 | 0.5.95 | 0.590 | 0.585 | 0.580 | 0.575 | 0.592 | 0.587 | 0.602 | 0.610 | 0.615 | 0.614 | |
S | 0.3317 | 0.3369 | 0.3339 | 0.3403 | 0.3213 | 0.3251 | 0.3296 | 0.3340 | 0.3354 | 0.3452 | 0.3445 | |
Refinery 13 | P1 | 0.632 | 0.629 | 0.635 | 0.625 | 0.615 | 0.629 | 0.638 | 0.698 | 0.674 | 0.589 | 0.650 |
P2 | 0.935 | 0.948 | 0.955 | 0.962 | 0.974 | 0.981 | 0.970 | 0.965 | 0.995 | 0.990 | 1.000 | |
P3 | 0.841 | 0.852 | 0.865 | 0.814 | 0.870 | 0.840 | 0.836 | 0.895 | 0.921 | 0.950 | 0.978 | |
S | 0.5699 | 0.5589 | 0.5779 | 0.6077 | 0.5923 | 0.6055 | 0.5911 | 0.6188 | 0.6119 | 0.6084 | 0.6278 | |
Refinery 14 | P1 | 0.550 | 0.557 | 0.560 | 0.565 | 0.572 | 0.563 | 0.571 | 0.559 | 0.574 | 0.576 | 0.580 |
P2 | 0.790 | 0.785 | 0.787 | 0.792 | 0.795 | 0.796 | 0.780 | 0.786 | 0.805 | 0.810 | 0.815 | |
P3 | 0.685 | 0.675 | 0.678 | 0.690 | 0.692 | 0.697 | 0.683 | 0.702 | 0.705 | 0.713 | 0.715 | |
S | 0.402 | 0.400 | 0.403 | 0.411 | 0.414 | 0.418 | 0.418 | 0.420 | 0.421 | 0.426 | 0.436 | |
Refinery 15 | P1 | 0.530 | 0.532 | 0.535 | 0.540 | 0.537 | 0.542 | 0.533 | 0.545 | 0.542 | 0.548 | 0.543 |
P2 | 0.765 | 0.762 | 0.768 | 0.771 | 0.773 | 0.766 | 0.775 | 0.778 | 0.764 | 0.782 | 0.785 | |
P3 | 0.682 | 0.673 | 0.675 | 0.685 | 0.671 | 0.689 | 0.678 | 0.684 | 0.689 | 0.710 | 0.713 | |
S | 0.4392 | 0.4430 | 0.3807 | 0.4632 | 0.4573 | 0.4433 | 0.4723 | 0.4684 | 0.4776 | 0.4739 | 0.4847 |
\(\alpha = 0\) | P1 | R13 ≻ R9 ≻ R8 ≻ R14 ≻ R15 ≻ R6 ≻ R5 ≻ R11 ≻ 12 ≻ R10 ≻ R4 ≻ R1 ≻ R2 ≻ R3 ≻ R7 |
P2 | R13 ≻ R9 ≻ R10 ≻ R8 ≻ R14 ≻ R4 ≻ R15 ≻ R1 ≻ R6 ≻ R2 ≻ R5 ≻ R3 ≻ R11 ≻ R7 ≻ R12 | |
P3 | R13 ≻ R9 ≻ R8 ≻ R14 ≻ R15 ≻ R6 ≻ R1 ≻ R4 ≻ R10 ≻ R5 ≻ R2 ≻ R3 ≻ R11 ≻ R7 ≻ R12 | |
S | R13 ≻ R9 ≻ R8 ≻ R4 ≻ R10 ≻ R1 ≻ R15 ≻ R14 ≻ R2 ≻ R3 ≻ R6 ≻ R7 ≻ R5 ≻ R11 ≻ R12 | |
\(\alpha = 0.1\) | P1 | R13 ≻ R9 ≻ R8 ≻ R14 ≻ R15 ≻ R6 ≻ R5 ≻ R11 ≻ R12 ≻ R10 ≻ R4 ≻ R1 ≻ R2 ≻ R3 ≻ R7 |
P2 | R13 ≻ R9 ≻ R10 ≻ R8 ≻ R14 ≻ R4 ≻ R15 ≻ R1 ≻ R6 ≻ R2 ≻ R5 ≻ R3 ≻ R11 ≻ R7 ≻ R12 | |
P3 | R13 ≻ R8 ≻ R9 ≻ R14 ≻ R15 ≻ R1 ≻ R6 ≻ R4 ≻ R10 ≻ R5 ≻ R2 ≻ R3 ≻ R11 ≻ R7 ≻ R12 | |
S | R13 ≻ R9 ≻ R8 ≻ R4 ≻ R10 ≻ R1 ≻ R14 ≻ R2 ≻ R15 ≻ R3 ≻ R6 ≻ R5 ≻ R7 ≻ R11 ≻ R12 | |
\(\alpha = 0.2\) | P1 | R13 ≻ R9 ≻ R8 ≻ R14 ≻ R15 ≻ R6 ≻ R5 ≻ R11 ≻ R12 ≻ R10 ≻ R4 ≻ R1 ≻ R2 ≻ R3 ≻ R7 |
P2 | R13 ≻ R9 ≻ R8 ≻ R10 ≻ R14 ≻ R4 ≻ R1 ≻ R15 ≻ R6 ≻ R2 ≻ R5 ≻ R3 ≻ R11 ≻ R7 ≻ R12 | |
P3 | R13 ≻ R9 ≻ R8 ≻ R14 ≻ R15 ≻ R1 ≻ R6 ≻ R5 ≻ R2 ≻ R3 ≻ R4 ≻ R10 ≻ R11 ≻ R7 ≻ R12 | |
S | R13 ≻ R9 ≻ R2 ≻ R8 ≻ R14 ≻ R15 ≻ R1 ≻ R3 ≻ R6 ≻ R4 ≻ R10 ≻ R7 ≻ R5 ≻ R11 ≻ R12 | |
\(\alpha = 0.3\) | P1 | R13 ≻ R9 ≻ R8 ≻ R14 ≻ R15 ≻ R6 ≻ R5 ≻ R11 ≻ R12 ≻ R10 ≻ R4 ≻ R1 ≻ R2 ≻ R3 ≻ R7 |
P2 | R13 ≻ R9 ≻ R10 ≻ R8 ≻ R4 ≻ R14 ≻ R1 ≻ R15 ≻ R6 ≻ R2 ≻ R3 ≻ R5 ≻ R11 ≻ R7 ≻ R12 | |
P3 | R13 ≻ R9 ≻ R8 ≻ R14 ≻ R15 ≻ R1 ≻ R2 ≻ R5 ≻ R6 ≻ R4 ≻ R10 ≻ R11 ≻ R3 ≻ R7 ≻ R12 | |
S | R13 ≻ R4 ≻ R9 ≻ R10 ≻ R2 ≻ R8 ≻ R14 ≻ R15 ≻ R3 ≻ R6 ≻ R1 ≻ R5 ≻ R11 ≻ R12 ≻ R7 | |
\(\alpha = 0.4\) | P1 | R13 ≻ R9 ≻ R8 ≻ R14 ≻ R15 ≻ R6 ≻ R5 ≻ R11 ≻ R12 ≻ R10 ≻ R4 ≻ R1 ≻ R2 ≻ R3 ≻ R7 |
P2 | R13 ≻ R9 ≻ R8 ≻ R10 ≻ R4 ≻ R14 ≻ R1 ≻ R15 ≻ R6 ≻ R2 ≻ R3 ≻ R5 ≻ R7 ≻ R11 ≻ R12 | |
P3 | R13 ≻ R8 ≻ R9 ≻ R14 ≻ R1 ≻ R15 ≻ R5 ≻ R4 ≻ R10 ≻ R2 ≻ R6 ≻ R11 ≻ R3 ≻ R7 ≻ R12 | |
S | R13 ≻ R9 ≻ R8 ≻ R10 ≻ R2 ≻ R1 ≻ R4 ≻ R14 ≻ R3 ≻ R15 ≻ R6 ≻ R7 ≻ R5 ≻ R11 ≻ R12 | |
\(\alpha = 0.5\) | P1 | R13 ≻ R9 ≻ R8 ≻ R14 ≻ R15 ≻ R6 ≻ R5 ≻ R11 ≻ R12 ≻ R10 ≻ R4 ≻ R1 ≻ R2 ≻ R3 ≻ R7 |
P2 | R13 ≻ R9 ≻ R8 ≻ R10 ≻ R4 ≻ R14 ≻ R1 ≻ R15 ≻ R2 ≻ R6 ≻ R3 ≻ R5 ≻ R7 ≻ R11 ≻ R12 | |
P3 | R13 ≻ R9 ≻ R8 ≻ R14 ≻ R15 ≻ R1 ≻ R2 ≻ R4 ≻ R10 ≻ R5 ≻ R6 ≻ R3 ≻ R11 ≻ R7 ≻ R12 | |
S | R13 ≻ R9 ≻ R8 ≻ R14 ≻ R4 ≻ R10 ≻ R2 ≻ R15 ≻ R1 ≻ R3 ≻ R6 ≻ R7 ≻ R5 ≻ R11 ≻ R12 | |
\(\alpha = 0.6\) | P1 | R13 ≻ R9 ≻ R8 ≻ R14 ≻ R15 ≻ R6 ≻ R5 ≻ R11 ≻ R12 ≻ R10 ≻ R4 ≻ R1 ≻ R2 ≻ R3 ≻ R7 |
P2 | R13 ≻ R9 ≻ R10 ≻ R8 ≻ R4 ≻ R1 ≻ R14 ≻ R15 ≻ R6 ≻ R3 ≻ R2 ≻ R5 ≻ R7 ≻ R11 ≻ R12 | |
P3 | R13 ≻ R9 ≻ R8 ≻ R14 ≻ R1 ≻ R15 ≻ R2 ≻ R4 ≻ R10 ≻ R5 ≻ R6 ≻ R3 ≻ R11 ≻ R7 ≻ R12 | |
S | R13 ≻ R9 ≻ R4 ≻ R10 ≻ R2 ≻ R8 ≻ R14 ≻ R1 ≻ R3 ≻ R15 ≻ R6 ≻ R7 ≻ R5 ≻ R11 ≻ R12 | |
\(\alpha = 0.7\) | P1 | R13 ≻ R9 ≻ R8 ≻ R14 ≻ R15 ≻ R6 ≻ R5 ≻ R11 ≻ R12 ≻ R10 ≻ R4 ≻ R1 ≻ R2 ≻ R3 ≻ R7 |
P2 | R13 ≻ R9 ≻ R8 ≻ R4 ≻ R1 ≻ R14 ≻ R15 ≻ R3 ≻ R2 ≻ R6 ≻ R7 ≻ R5 ≻ R10 ≻ R11 ≻ R12 | |
P3 | R13 ≻ R9 ≻ R8 ≻ R14 ≻ R1 ≻ R15 ≻ R5 ≻ R6 ≻ R4 ≻ R10 ≻ R3 ≻ R2 ≻ R7 ≻ R11 ≻ R12 | |
S | R13 ≻ R9 ≻ R8 ≻ R14 ≻ R1 ≻ R4 ≻ R2 ≻ R10 ≻ R15 ≻ R3 ≻ R6 ≻ R7 ≻ R5 ≻ R11 ≻ R12 | |
\(\alpha = 0.8\) | P1 | R13 ≻ R9 ≻ R8 ≻ R14 ≻ R15 ≻ R6 ≻ R5 ≻ R11 ≻ R12 ≻ R10 ≻ R4 ≻ R1 ≻ R2 ≻ R3 ≻ R7 |
P2 | R13 ≻ R9 ≻ R10 ≻ R4 ≻ R8 ≻ R1 ≻ R14 ≻ R2 ≻ R3 ≻ R15 ≻ R6 ≻ R7 ≻ R5 ≻ R11 ≻ R12 | |
P3 | R13 ≻ R9 ≻ R8 ≻ R14 ≻ R15 ≻ R6 ≻ R4 ≻ R10 ≻ R1 ≻ R5 ≻ R3 ≻ R2 ≻ R7 ≻ R11 ≻ R12 | |
S | R13 ≻ R9 ≻ R10 ≻ R8 ≻ R4 ≻ R1 ≻ R14 ≻ R2 ≻ R15 ≻ R3 ≻ R6 ≻ R7 ≻ R5 ≻ R11 ≻ R12 | |
\(\alpha = 0.9\) | P1 | R9 ≻ R13 ≻ R8 ≻ R14 ≻ R15 ≻ R6 ≻ R5 ≻ R11 ≻ R12 ≻ R10 ≻ R4 ≻ R1 ≻ R2 ≻ R3 ≻ R7 |
P2 | R13 ≻ R9 ≻ R10 ≻ R8 ≻ R4 ≻ R1 ≻ R14 ≻ R15 ≻ R2 ≻ R6 ≻ R3 ≻ R5 ≻ R7 ≻ R11 ≻ R12 | |
P3 | R13 ≻ R9 ≻ R8 ≻ R14 ≻ R15 ≻ R1 ≻ R2 ≻ R4 ≻ R10 ≻ R6 ≻ R5 ≻ R3 ≻ R7 ≻ R11 ≻ R12 | |
S | R13 ≻ R9 ≻ R4 ≻ R10 ≻ R8 ≻ R9 ≻ R15 ≻ R2 ≻ R14 ≻ R3 ≻ R6 ≻ R7 ≻ R5 ≻ R11 ≻ R12 | |
\(\alpha = 1\) | P1 | R13 ≻ R9 ≻ R8 ≻ R14 ≻ R15 ≻ R6 ≻ R5 ≻ R11 ≻ R12 ≻ R10 ≻ R4 ≻ R1 ≻ R2 ≻ R3 ≻ R7 |
P2 | R13 ≻ R9 ≻ R10 ≻ R8 ≻ R1 ≻ R4 ≻ R14 ≻ R3 ≻ R15 ≻ R2 ≻ R6 ≻ R7 ≻ R5 ≻ R11 ≻ R12 | |
P3 | R13 ≻ R9 ≻ R8 ≻ R14 ≻ R15 ≻ R1 ≻ R2 ≻ R4 ≻ R10 ≻ R5 ≻ R6 ≻ R3 ≻ R7 ≻ R11 ≻ R12 | |
S | R13 ≻ R9 ≻ R8 ≻ R10 ≻ R4 ≻ R2 ≻ R1 ≻ R15 ≻ R14 ≻ R3 ≻ R6 ≻ R7 ≻ R5 ≻ R11 ≻ R12 |
Conclusions
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The proposed approach for solving fuzzy relational NDEA model increases the number of constraints significantly rather than the primary fuzzy DEA model. This issue can increase the complexity of the model for large-scale real-world applications.
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The proposed algorithm gives the membership functions of the fuzzy efficiency scores numerically. Thus, the proposed algorithm cannot be useful for situations in which decision makers needs the exact form of membership functions for the fuzzy efficiency scores.
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In addition to desirable outputs, undesirable outputs often arise in the production process in real-world applications, especially when evaluating environmental performance. Thus, the performance evaluation becomes more complex when desirable outputs and undesirable outputs simultaneously exist in the system.
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Development of a new fuzzy approach that not only decreases the number of constraints but also provides the exact form of membership functions is left to the next study.
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Development of the proposed fuzzy relational NDEA model in the presence of undesirable outputs could be an interesting topic for future research.