Allocating Benefits Due to Shared Resources Using Shapley Value and Nucleolus in Dynamic Network Data Envelopment Analysis
Abstract
:1. Introduction
2. Literature Review
2.1. Dynamic DEA with Network Structure
2.2. Resource Sharing
2.3. Cooperative Games in the DEA Field
Paper | Area | Objective | Game Approach | DEA Model | DMU | Inputs | Outputs | Link/ Stages |
---|---|---|---|---|---|---|---|---|
[61] | - | Allocating or imputing benefits | Shapley value and Nucleolus | CCR | - | - | - | - |
[65] | - | Resource allocation | Shapley value and Nucleolus | CCR | - | - | - | - |
[66] | - | Propose a cross-efficiency game | Shapley value | Cross efficiency | 5 | 3 | 2 | - |
[67] | - | Importance of variables in DEA | Shapley value | Radial DEA | 8 | 6 | 2 | - |
[68] | - | Investigate the benefits of sharing data among DMUs | Shapley value, Nucleolus, and τ-value | Cost DEA | 12 | 2 | 2 | - |
[69] | - | Stable payoff allocation | Nucleolus | DEA production game | 3 | 2 | 2 | - |
[70] | Energy | Increase DEA discrimination | Shapley value | MODEA | 20 | 13 | 3 | - |
[63] | - | Resource allocation | Shapley value | CCR | 12 | 3 | 2 | - |
[71] | Banks | Efficient DMU evaluation | Shapley value | Super efficiency | 14 | 3 | 3 | - |
[3] | Paper Industry and Ports | Resource allocation | Shapley value | MILP DEA VRS | 8/28 | 4 | 2/3 | - |
[72] | Banks | Efficient DMU evaluation | Shapley value | CCR | 14 | 3 | 2 | - |
[59] | Energy | Resource allocation | Shapley value and Nucleolus | Cross efficiency | 4 | 4 | 3 | - |
[62] | Banks | Resource allocation | Shapley value | Cross efficiency | 18 | 3 | 3 | - |
[73] | Health | Fully ranking DMUs | Core and Shapley value | Cross efficiency | 288 | 3 | 4 | - |
[74] | Environment | Cost savings allocation | Shapley value | CCR | 6 | 3 | 2 | - |
[75] | Transportation | Fully ranking DMUs | Shapley Value | Cross efficiency | 9 | 7 | 2 | - |
[76] | Manufacturing | Resource allocation | Nucleolus | Cross efficiency | 10 | 4 | 2 | - |
[77] | Energy | Efficient DMU evaluation | Shapley value | Cross efficiency | 31 | 5 | 4 | - |
[2] | - | Resource allocation | Shapley value | NDEA | 10 | 3 | 1 | 2/3 |
[78] | Energy/Environment | Increase DEA discrimination | Shapley value | Cross-efficiency DEA Game | 17 | 1 | 13 | - |
[79] | Environment | Resource allocation/target setting | Nucleolus | DDF DEA | 31 | 3 | 2 | - |
[80] | Banks | Cost allocation | Shapley value | CCR | 5 | 2 | 1 | - |
[60] | Banks | Resource/cost allocation | Nucleolus | NDEA | 27 | 3 | 2 | 3/2 |
[81] | Cities development | Composite indicator construction | Shapley value | DEA Game | 13 | 1 | 68 | - |
[82] | Banks | Efficient DMU evaluation | Shapley value | CCR | 14 | 3 | 2 | - |
[83] | Environment | Resource allocation/target setting | Shapley value | Zero-sum DEA | 9 | 2 | 1 | - |
[84] | Logistics/Environment | Resource allocation | Shapley value | CRS, VRS, and MRS DEA | 23 | 5 | 7 | - |
[85] | Supply chain | Profit allocation | Shapley value and Nucleolus | Double level NDEA | 15 | 4 | 4 | 4 |
[54] | Transportation | Resource allocation | Core, Shapley value, and Nucleolus | Parallel DEA | 8 | 5 | 3 | 3 |
[55] | Hotel | Resource allocation | Shapley value | Parallel DEA | 7 | 3 | 2 | 2 |
[86] | Mineral resources | Community partition | Shapley value | Revenue DEA | 31 | 2 | 3 | - |
Torres and Ramos | - | Resource sharing and additional profit allocation | Shapley value and Nucleolus | DNDEA | 10 | 6 | 3 | 3/3 |
3. Resource Sharing in a Dynamic DEA Model with a Three-Stage Network
3.1. Analysis of the Pre-Collaboration between Stages
3.2. Post-Collaboration and Coalitions
3.3. Payoff Allocation Using Shapley Value and Nucleolus
4. Numerical Example
- Free carry-overs;
- Fixed carry-overs.
4.1. Characteristic Functions
4.2. Payoff Allocation with Shapley Value and Nucleolus
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Indexes | |
Index of the jth DMU; | |
Index of the tth period; | |
Index of the lth stage; | |
Index of the ith shared input between stages; | |
Index of the mth specific input of stage ; | |
Index of the kth output produced by stage ; | |
Index of the dth carry-over connecting stage l between periods; | |
Parameters | |
/// | The unit price of the ith shared input in period t; the mth specific input of stage in period t; the kth output of the stage l in period t; the dth carry-over connecting stage l between consecutive periods t and t + 1; |
/// | The ith shared input of DMU j for stage in period ; the specif input of DMU j for stage in period ; the kth output of DMU j for stage ; the dth carry-over connecting stage l between consecutive periods; |
Coalition | |
Variables | |
/ | Index of multiplier variable corresponding to the stage l in period t of DMU j before/after resource sharing; |
/// | The index of ith shared input of DMU j for stage in period ; the specif input of DMU j for stage in period ; the kth output of DMU j for stage ; the dth carry-over connecting stage l between consecutive periods in the optimal situation before resource sharing; |
/// | The index of ith shared input of DMU j for stage in period ; the specif input of DMU j for stage in period ; the kth output of DMU j for stage ; the dth carry-over connecting stage l between consecutive periods in the optimal situation after resource sharing. |
t = 1 | DMU | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Unit Price |
Stage 1 | I = 2 | 18.49 | 11.42 | 11.18 | 5.56 | 1.86 | 13.74 | 5.48 | 21.79 | 9.06 | 3.01 | 9.99 |
1.53 | 3.91 | 11.05 | 2.29 | 21.63 | 1.36 | 1.02 | 1.15 | 7.10 | 9.11 | 8.58 | ||
M = 1 | 15.95 | 7.26 | 10.11 | 2.85 | 8.44 | 1.53 | 12.90 | 3.54 | 6.91 | 12.34 | 1.56 | |
K = 1 | 23.23 | 17.36 | 26.98 | 1.28 | 12.93 | 13.29 | 10.68 | 20.68 | 20.71 | 8.59 | 0 | |
C = 1 | 18.91 | 21.81 | 13.70 | 21.54 | 19.58 | 20.87 | 11.97 | 10.30 | 27.70 | 21.84 | 10.8 | |
Stage 2 | I = 2 | 1.68 | 2.64 | 11.29 | 3.97 | 18.63 | 2.17 | 7.35 | 1.23 | 2.81 | 2.61 | 9.99 |
6.58 | 12.00 | 2.22 | 8.39 | 4.25 | 10.06 | 14.52 | 1.23 | 16.89 | 9.62 | 8.58 | ||
M = 1 | 1.02 | 2.13 | 3.87 | 2.35 | 1.64 | 1.12 | 2.06 | 1.81 | 1.32 | 1.49 | 9.12 | |
K = 1 | 14.62 | 16.91 | 18.63 | 6.62 | 16.96 | 12.85 | 16.64 | 11.94 | 20.01 | 13.57 | 0 | |
C = 1 | 28.51 | 13.77 | 7.70 | 22.37 | 15.82 | 22.53 | 3.16 | 23.58 | 14.63 | 21.97 | 6.29 | |
Stage 3 | I = 2 | 2.13 | 17.10 | 1.28 | 5.40 | 3.01 | 19.08 | 4.17 | 1.01 | 4.22 | 6.72 | 9.99 |
10.10 | 3.93 | 9.40 | 1.44 | 2.99 | 2.20 | 1.56 | 18.11 | 1.02 | 2.35 | 8.58 | ||
M = 1 | 4.10 | 1.15 | 3.42 | 3.26 | 1.46 | 2.32 | 2.29 | 1.09 | 1.55 | 2.41 | 2.52 | |
K = 1 | 27.49 | 30.18 | 30.91 | 16.38 | 29.55 | 25.26 | 26.23 | 29.75 | 26.99 | 25.96 | 8.83 | |
C = 1 | 1.04 | 23.46 | 12.42 | 3.32 | 12.67 | 29.62 | 22.34 | 13.30 | 6.88 | 24.74 | 8.73 | |
t = 2 | DMU | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Unit Price |
Stage 1 | I = 2 | 11.94 | 7.29 | 25.28 | 2.09 | 20.78 | 26.41 | 22.70 | 8.59 | 10.03 | 26.50 | 8.8 |
4.89 | 30.69 | 6.13 | 2.58 | 22.67 | 12.02 | 7.35 | 8.25 | 4.61 | 27.63 | 8.39 | ||
M = 1 | 17.57 | 14.04 | 28.81 | 29.01 | 22.96 | 28.59 | 3.72 | 11.87 | 30.48 | 20.35 | 1.84 | |
K = 1 | 8.39 | 4.98 | 9.67 | 1.18 | 17.5 | 13.86 | 25.64 | 18.83 | 26.13 | 1.11 | 0 | |
C = 1 | 8.52 | 23.22 | 26.73 | 26.71 | 5.12 | 1.91 | 3.90 | 21.40 | 16.44 | 20.52 | 5.05 | |
Stage 2 | I = 2 | 21.30 | 13.26 | 16.05 | 29.10 | 19.11 | 7.24 | 30.54 | 30.34 | 16.93 | 28.41 | 8.8 |
30.01 | 15.08 | 14.53 | 24.70 | 29.58 | 21.23 | 19.42 | 11.96 | 27.53 | 6.30 | 8.39 | ||
M = 1 | 4.16 | 25.72 | 3.49 | 28.40 | 4.18 | 18.73 | 23.48 | 2.16 | 12.55 | 8.86 | 7.32 | |
K = 1 | 15.83 | 15.86 | 9.14 | 8.66 | 21.51 | 25.13 | 7.78 | 13.62 | 6.59 | 26.67 | 0 | |
C = 1 | 24.31 | 25.60 | 10.23 | 22.51 | 14.51 | 24.27 | 4.72 | 23.14 | 2.35 | 7.28 | 4.20 | |
Stage 3 | I = 2 | 30.03 | 22.52 | 19.62 | 4.71 | 14.20 | 20.77 | 26.30 | 24.07 | 2.91 | 28.97 | 8.8 |
30.48 | 13.06 | 17.31 | 22.21 | 11.21 | 27.25 | 25.71 | 23.98 | 8.05 | 18.67 | 8.39 | ||
M = 1 | 4.65 | 5.02 | 27.26 | 21.91 | 12.51 | 18.44 | 4.94 | 25.56 | 13.19 | 28.10 | 1.95 | |
K = 1 | 14.43 | 26.70 | 29.43 | 25.12 | 26.82 | 11.04 | 1.15 | 29.26 | 26.11 | 16.95 | 10.91 | |
C = 1 | 15.00 | 20.76 | 28.94 | 16.62 | 8.74 | 5.63 | 7.00 | 10.51 | 20.76 | 13.26 | 9.69 |
Case 1: Fixed Carry-Overs | ||||||||||
Characteristic Function | DMU1 | DMU2 | DMU3 | DMU4 | DMU5 | DMU6 | DMU7 | DMU8 | DMU9 | DMU10 |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |
162.29 | 204.46 | 228.36 | 433.21 | 766.91 | 804.75 | 307.40 | 0.00 | 200.69 | 0.00 | |
0.00 | 0.00 | 0.00 | 233.16 | 766.91 | 0.00 | 307.40 | 0.04 | 309.86 | 0.00 | |
0.00 | 389.88 | 0.00 | 254.81 | 766.91 | 850.96 | 307.40 | 0.00 | 168.36 | 0.00 | |
221.21 | 389.88 | 228.36 | 460.67 | 766.91 | 850.96 | 307.40 | 76.38 | 350.23 | 0.00 | |
162.29 | 204.46 | 228.36 | 433.21 | 766.91 | 804.75 | 307.40 | 0.00 | 200.69 | 0.00 | |
0.00 | 0.00 | 0.00 | 233.16 | 766.91 | 0.00 | 307.40 | 0.04 | 309.86 | 0.00 | |
0.00 | 389.88 | 0.00 | 254.81 | 766.91 | 850.96 | 307.40 | 0.00 | 168.36 | 0.00 | |
221.21 | 389.88 | 228.36 | 460.67 | 766.91 | 850.96 | 307.40 | 76.38 | 350.23 | 0.00 | |
58.92 | 185.42 | 0.00 | 27.46 | 0.00 | 46.20 | 0.00 | 76.38 | 149.54 | 0.00 | |
221.21 | 0.00 | 228.36 | 205.86 | 0.00 | 0.00 | 0.00 | 76.38 | 181.87 | 0.00 | |
221.21 | 389.88 | 228.36 | 227.51 | 0.00 | 850.96 | 0.00 | 76.34 | 40.38 | 0.00 | |
Case 2: Free Carry-Overs | ||||||||||
Characteristic Function | DMU1 | DMU2 | DMU3 | DMU4 | DMU5 | DMU6 | DMU7 | DMU8 | DMU9 | DMU10 |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |
125.97 | 259.95 | 346.04 | 282.81 | 295.22 | 138.22 | 820.61 | 408.94 | 93.34 | 20.04 | |
0.00 | 1.10 | 477.91 | 0.00 | 356.45 | 138.22 | 0.00 | 408.94 | 170.68 | 42.04 | |
0.00 | 1.10 | 491.43 | 0.00 | 356.45 | 138.22 | 862.95 | 408.94 | 171.01 | 20.04 | |
125.97 | 259.95 | 491.43 | 282.81 | 532.88 | 138.22 | 862.95 | 408.94 | 171.01 | 45.73 | |
125.97 | 259.95 | 346.04 | 282.81 | 295.22 | 138.22 | 820.61 | 408.94 | 93.34 | 20.04 | |
0.00 | 1.10 | 477.91 | 0.00 | 356.45 | 138.22 | 0.00 | 408.94 | 170.68 | 42.04 | |
0.00 | 1.10 | 491.43 | 0.00 | 356.45 | 138.22 | 862.95 | 408.94 | 171.01 | 20.04 | |
125.97 | 259.95 | 491.43 | 282.81 | 532.88 | 138.22 | 862.95 | 408.94 | 171.01 | 45.73 | |
0.00 | 0.00 | 145.39 | 0.00 | 237.65 | 0.00 | 42.34 | 0.00 | 77.67 | 25.69 | |
125.97 | 258.85 | 0.00 | 282.81 | 176.43 | 0.00 | 0.00 | 0.00 | 0.00 | 25.69 | |
125.97 | 258.85 | 13.53 | 282.81 | 176.43 | 0.00 | 862.95 | 0.00 | 0.34 | 3.69 |
Case 1: Fixed Carry-Overs | ||||||||||
Shapley Value | DMU 1 | DMU 2 | DMU 3 | DMU 4 | DMU 5 | DMU 6 | DMU 7 | DMU 8 | DMU 9 | DMU 10 |
Stage 1 | 100.79 | 34.08 | 114.18 | 179.68 | 255.64 | 134.13 | 102.47 | 25.47 | 145.72 | 0.00 |
Stage 2 | 100.79 | 229.02 | 114.18 | 190.51 | 255.64 | 559.60 | 102.47 | 25.45 | 74.97 | 0.00 |
Stage 3 | 19.64 | 126.79 | 0.00 | 90.48 | 255.64 | 157.23 | 102.47 | 25.47 | 129.55 | 0.00 |
Case 2: Free Carry-Overs | ||||||||||
Shapley Value | DMU 1 | DMU 2 | DMU 3 | DMU 4 | DMU 5 | DMU 6 | DMU 7 | DMU 8 | DMU 9 | DMU 10 |
Stage 1 | 62.98 | 129.79 | 137.33 | 141.40 | 167.42 | 46.07 | 136.77 | 136.31 | 44.00 | 18.91 |
Stage 2 | 62.98 | 129.79 | 144.09 | 141.40 | 167.42 | 46.07 | 568.24 | 136.31 | 44.17 | 7.91 |
Stage 3 | 0.00 | 0.37 | 210.02 | 0.00 | 198.03 | 46.07 | 157.94 | 136.31 | 82.84 | 18.91 |
Case 1: Fixed Carry-Overs | ||||||||||
Nucleolus | DMU 1 | DMU 2 | DMU 3 | DMU 4 | DMU 5 | DMU 6 | DMU 7 | DMU 8 | DMU 9 | DMU 10 |
Stage 1 | 95.88 | 0.00 | 114.18 | 205.81 | 255.64 | 0.00 | 102.47 | 25.47 | 174.69 | 0.00 |
Stage 2 | 95.88 | 297.17 | 114.18 | 227.46 | 255.64 | 827.86 | 102.47 | 25.47 | 33.19 | 0.00 |
Stage 3 | 29.46 | 92.71 | 0.00 | 27.41 | 255.64 | 23.11 | 102.47 | 25.47 | 142.36 | 0.00 |
Case 2: Free Carry-Overs | ||||||||||
Nucleolus | DMU 1 | DMU 2 | DMU 3 | DMU 4 | DMU 5 | DMU 6 | DMU 7 | DMU 8 | DMU 9 | DMU 10 |
Stage 1 | 62.99 | 129.98 | 110.84 | 141.41 | 157.22 | 46.07 | 0.00 | 136.31 | 31.00 | 21.94 |
Stage 2 | 62.99 | 129.98 | 124.36 | 141.41 | 157.22 | 46.07 | 841.78 | 136.31 | 31.33 | 1.85 |
Stage 3 | 0.00 | 0.00 | 256.23 | 0.00 | 218.45 | 46.07 | 21.17 | 136.31 | 108.67 | 21.94 |
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Torres, L.; Ramos, F.S. Allocating Benefits Due to Shared Resources Using Shapley Value and Nucleolus in Dynamic Network Data Envelopment Analysis. Mathematics 2024, 12, 698. https://doi.org/10.3390/math12050698
Torres L, Ramos FS. Allocating Benefits Due to Shared Resources Using Shapley Value and Nucleolus in Dynamic Network Data Envelopment Analysis. Mathematics. 2024; 12(5):698. https://doi.org/10.3390/math12050698
Chicago/Turabian StyleTorres, Lívia, and Francisco S. Ramos. 2024. "Allocating Benefits Due to Shared Resources Using Shapley Value and Nucleolus in Dynamic Network Data Envelopment Analysis" Mathematics 12, no. 5: 698. https://doi.org/10.3390/math12050698
APA StyleTorres, L., & Ramos, F. S. (2024). Allocating Benefits Due to Shared Resources Using Shapley Value and Nucleolus in Dynamic Network Data Envelopment Analysis. Mathematics, 12(5), 698. https://doi.org/10.3390/math12050698