Table 3 Payoffs for the benefit type criteria in six scenarios
Case | Relation | Benefit type | Cost type | ||
|---|---|---|---|---|---|
Loss | Gain | Loss | Gain | ||
1 | \(C_{{}}^{H} < R_{{}}^{L}\) | \(0.5(C_{{}}^{L} + C_{{}}^{H} ) - R_{{}}^{L}\) | 0 | \(R_{{}}^{L} - 0.5(C_{{}}^{L} + C_{{}}^{H} )\) | 0 |
2 | \(R_{{}}^{H} < C_{{}}^{L}\) | 0 | \(0.5(C_{{}}^{L} + C_{{}}^{H} ) - R_{{}}^{H}\) | 0 | \(R_{{}}^{H} - 0.5(C_{{}}^{L} + C_{{}}^{H} )\) |
3 | \(C_{{}}^{L} < R_{{}}^{L} \le C_{{}}^{H} < R_{{}}^{H}\) | \(0.5(C_{{}}^{L} - R_{{}}^{L} )\) | 0 | \(0.5(R_{{}}^{L} - C_{{}}^{L} )\) | 0 |
4 | \(R_{{}}^{L} < C_{{}}^{L} \le R_{{}}^{H} < C_{{}}^{H}\) | 0 | \(0.5(C_{{}}^{H} - R_{{}}^{H} )\) | 0 | \(0.5(R_{{}}^{H} - C_{{}}^{H} )\) |
5 | \(C_{{}}^{L} < R_{{}}^{L} < R_{{}}^{H} < C_{{}}^{H}\) | \(0.5(C_{{}}^{L} - R_{{}}^{L} )\) | \(0.5(C_{{}}^{H} - R_{{}}^{H} )\) | \(0.5(R_{{}}^{L} - C_{{}}^{L} )\) | \(0.5(R_{{}}^{H} - C_{{}}^{H} )\) |
6 | \(R_{{}}^{L} \le C_{{}}^{L} < C_{{}}^{H} \le R_{{}}^{H}\) | 0 | 0 | 0 | 0 |