# American Institute of Mathematical Sciences

March  2020, 16(2): 633-651. doi: 10.3934/jimo.2018171

## A new method for ranking decision making units using common set of weights: A developed criterion

 Department of Mathematics, Faculty of Sciences, University of Qom, Alghadir Bld., Postal code:3716146611, Qom, Iran

* Corresponding author: Gholam Hassan Shirdel

Received  December 2016 Revised  June 2017 Published  October 2018

In this paper we have developed a new model by altering Liu and Peng's approach [20] toward ranking method using CSW. In fact, we have adopted a new criterion which is stronger in terms of maximizing efficiencies. After showing advantages of our model theoretically and illustrating it geometrically, two examples demonstrated how the proposed method is practically more capable.

Citation: Gholam Hassan Shirdel, Somayeh Ramezani-Tarkhorani. A new method for ranking decision making units using common set of weights: A developed criterion. Journal of Industrial & Management Optimization, 2020, 16 (2) : 633-651. doi: 10.3934/jimo.2018171
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##### References:
$0<y_P<x_P, \Delta_P = x_p-y_p ,m_p = {y_p \over x_p} , o<m_{p_1}<m_P<m_{p_2}$
The interior points of the area $OQ_1P$, $R_1$, have less amount of $\Delta$ than $P$ while slopes of crossing lines from the origin and these points are less than $m_P$
Slopes of crossing lines from the origin and the points located in the areas $R_1$ and $R_2$ are less than $m_P$
Range of slopes of the lines crossing the origin and the points located in $R_3$ is the same as that for the the points located in the area enclosed by the segments $OP$, $OQ_6$ and $Q_6P$.
Slope of the line crossing the origin and $P^{(U_1,V_1)}_o$ is maximum.
Two pairs of DMUs with the same efficiency scores.
Input and Output data of DMUs
 $DMU_j$ $x_{1j}$ $x_{2j}$ $x_{3j}$ $x_{4j}$ $y_{1j}$ $y_{2j}$ $DMU_1$ 995 6205 1375 2629 4127 1678 $DMU_2$ 917 5898 1379 2047 3721 1277 $DMU_3$ 3178 10049 3615 3511 2706 2051 $DMU_4$ 813 5833 1124 1730 2176 1538 $DMU_5$ 1236 8639 2486 4990 5220 2042 $DMU_6$ 1146 7610 1600 3589 3517 1856 $DMU_7$ 705 5600 1557 3623 2352 2060 $DMU_8$ 2871 11524 2880 2452 1755 1664 $DMU_9$ 1098 8998 1730 2823 4412 2334 $DMU_{10}$ 2032 9383 2421 4454 5386 2080 $DMU_{11}$ 1414 10468 2140 3649 5735 2691 $DMU_{12}$ 1967 11260 2759 3178 6079 2804 $DMU_{13}$ 1851 9880 2335 4570 5893 2495 $DMU_{14}$ 3100 15649 5487 2940 5248 3692 $DMU_{15}$ 5016 18010 4008 3567 7800 4852 $DMU_{16}$ 1924 12682 2490 2975 6040 3396
 $DMU_j$ $x_{1j}$ $x_{2j}$ $x_{3j}$ $x_{4j}$ $y_{1j}$ $y_{2j}$ $DMU_1$ 995 6205 1375 2629 4127 1678 $DMU_2$ 917 5898 1379 2047 3721 1277 $DMU_3$ 3178 10049 3615 3511 2706 2051 $DMU_4$ 813 5833 1124 1730 2176 1538 $DMU_5$ 1236 8639 2486 4990 5220 2042 $DMU_6$ 1146 7610 1600 3589 3517 1856 $DMU_7$ 705 5600 1557 3623 2352 2060 $DMU_8$ 2871 11524 2880 2452 1755 1664 $DMU_9$ 1098 8998 1730 2823 4412 2334 $DMU_{10}$ 2032 9383 2421 4454 5386 2080 $DMU_{11}$ 1414 10468 2140 3649 5735 2691 $DMU_{12}$ 1967 11260 2759 3178 6079 2804 $DMU_{13}$ 1851 9880 2335 4570 5893 2495 $DMU_{14}$ 3100 15649 5487 2940 5248 3692 $DMU_{15}$ 5016 18010 4008 3567 7800 4852 $DMU_{16}$ 1924 12682 2490 2975 6040 3396
The generated common set of weights
 $v^*_1$ $v^*_2$ $v^*_3$ $v^*_4$ $u^*_1$ $u^*_2$ Our method 0.000001 0.181081 0.000001 0.124392 0.116466 0.578059 Liu and Peng's method [20] 0.000001 0.000001 0.000001 7.50E-07 8.70E- 07 0.000004
 $v^*_1$ $v^*_2$ $v^*_3$ $v^*_4$ $u^*_1$ $u^*_2$ Our method 0.000001 0.181081 0.000001 0.124392 0.116466 0.578059 Liu and Peng's method [20] 0.000001 0.000001 0.000001 7.50E-07 8.70E- 07 0.000004
The efficiencies
 $DMU_j$ $E^*_j$ CWA-Efficiency CCR-Efficiency $DMU_1$ 1 1 1 $DMU_2$ 0.885761 0.877007 1 $DMU_3$ 0.665101 0.55716 0.690363 $DMU_4$ 0.89857 0.911488 1 $DMU_5$ 0.818436 0.807723 1 $DMU_6$ 0.812557 0.823997 0.881091 $DMU_7$ 1 1 1 $DMU_8$ 0.48762 0.440469 0.555791 $DMU_9$ 0.940676 0.968986 1 $DMU_{10}$ 0.812047 0.774854 0.863042 $DMU_{11}$ 0.94638 0.963253 0.996068 $DMU_{12}$ 0.956693 0.920591 1 $DMU_{13}$ 0.90288 0.884299 0.915511 $DMU_{14}$ 0.858084 0.75116 1 $DMU_{15}$ 0.960084 0.867478 1 $DMU_{16}$ 1 1 1
 $DMU_j$ $E^*_j$ CWA-Efficiency CCR-Efficiency $DMU_1$ 1 1 1 $DMU_2$ 0.885761 0.877007 1 $DMU_3$ 0.665101 0.55716 0.690363 $DMU_4$ 0.89857 0.911488 1 $DMU_5$ 0.818436 0.807723 1 $DMU_6$ 0.812557 0.823997 0.881091 $DMU_7$ 1 1 1 $DMU_8$ 0.48762 0.440469 0.555791 $DMU_9$ 0.940676 0.968986 1 $DMU_{10}$ 0.812047 0.774854 0.863042 $DMU_{11}$ 0.94638 0.963253 0.996068 $DMU_{12}$ 0.956693 0.920591 1 $DMU_{13}$ 0.90288 0.884299 0.915511 $DMU_{14}$ 0.858084 0.75116 1 $DMU_{15}$ 0.960084 0.867478 1 $DMU_{16}$ 1 1 1
The average of the efficiencies in each method and their ratio
 Our method Liu and Peng's method [20] The ratio 0.871556 0.846779 1.02926
 Our method Liu and Peng's method [20] The ratio 0.871556 0.846779 1.02926
The ranking scores
 $DMU_j$ Our method: Input-oriented approach Our method: Output-oriented approach CWA-ranking CCR $DMU_1$ 1 3 2 1 $DMU_2$ 10 10 9 1 $DMU_3$ 15 15 15 6 $DMU_4$ 9 9 7 1 $DMU_5$ 12 12 12 1 $DMU_6$ 13 13 11 4 $DMU_7$ 2 2 3 1 $DMU_8$ 16 16 16 7 $DMU_9$ 7 7 4 1 $DMU_{10}$ 14 14 13 5 $DMU_{11}$ 6 6 5 2 $DMU_{12}$ 5 5 6 1 $DMU_{13}$ 8 8 8 3 $DMU_{14}$ 11 11 14 1 $DMU_{15}$ 4 4 10 1 $DMU_{16}$ 3 1 1 1
 $DMU_j$ Our method: Input-oriented approach Our method: Output-oriented approach CWA-ranking CCR $DMU_1$ 1 3 2 1 $DMU_2$ 10 10 9 1 $DMU_3$ 15 15 15 6 $DMU_4$ 9 9 7 1 $DMU_5$ 12 12 12 1 $DMU_6$ 13 13 11 4 $DMU_7$ 2 2 3 1 $DMU_8$ 16 16 16 7 $DMU_9$ 7 7 4 1 $DMU_{10}$ 14 14 13 5 $DMU_{11}$ 6 6 5 2 $DMU_{12}$ 5 5 6 1 $DMU_{13}$ 8 8 8 3 $DMU_{14}$ 11 11 14 1 $DMU_{15}$ 4 4 10 1 $DMU_{16}$ 3 1 1 1
The averages of the obtained efficiencies and their ratio in each case
 Size of the sample Our method CWA-Efficiencies The ratio 6 0.916128 0.906961 1.010107 12 0.539086 0.539082 1.000006 25 0.689937 0.148176 4.656189 50 0.550724 0.553320 0.995309 100 0.639243 0.471678 1.355254 200 0.582310 0.580711 1.002753
 Size of the sample Our method CWA-Efficiencies The ratio 6 0.916128 0.906961 1.010107 12 0.539086 0.539082 1.000006 25 0.689937 0.148176 4.656189 50 0.550724 0.553320 0.995309 100 0.639243 0.471678 1.355254 200 0.582310 0.580711 1.002753