Table 7 Compromised solutions with respect to improvement desired in various objective at confidence level \(\alpha =0.9\) with different shape parameter
Case | Obj. function | Bounds | \(\lambda\) | Objective values | Solution variables |
|---|---|---|---|---|---|
\(Z_{1}, Z_{2}, Z_{3}\) | \(x_{ij}\) | ||||
Shape parameter: (−5, −1, −2) | |||||
Aspiration level: (0.8, 0.85, 0.7) | |||||
1 | Cost | \(22.2 \le z_{11} \le 40.9\), | 0.8578 | (33.1, 34, 35.5), | \(x_{11}, x_{13}, x_{36}, x_{44}, x_{55}, x_{62}\) |
\(23 \le z_{12} \le 42\), | (42.7, 44, 45.6), | ||||
\(24 \le z_{13} \le 43.7\) | (19, 20, 21.2) | ||||
2 | Time | \(28 \le z_{21} \le 36.9\), | 0.7574 | (46.7, 48, 49.8), | \(x_{14}, x_{31}, x_{36}, x_{52}, x_{55}, x_{62}\) |
\(29 \le z_{22} \le 38\), | (28, 29, 30.4), | ||||
\(30.3 \le z_{23} \le 39.5\) | (25, 26, 27.1) | ||||
3 | Quality | \(11.1 \le z_{31} \le 15\), | 0.8681 | (40, 41, 42.7), | \(x_{13}, x_{14}, x_{31}, x_{46}, x_{55}, x_{62}\) |
\(12 \le z_{32} \le 16\), | (41.9, 43, 44.6), | ||||
\(13.2 \le z_{33} \le 17.2\) | (11.1, 12, 13.2) | ||||
Aspiration level: (0.9, 0.7, 0.8) | |||||
1 | Cost | \(22.2 \le z_{11} \le 36.1\), | 0.7322 | (29.3, 30, 31.3), | \(x_{14}, x_{23}, x_{35}, x_{36}, x_{51}, x_{62}\) |
\(23 \le z_{12} \le 37\), | (53.7, 55, 56.8), | ||||
\(24 \le z_{13} \le 38.6\) | (20.9, 22, 23.2) | ||||
2 | Time | \(28 \le z_{21} \le 41.8\), | 0.8029 | (41.8, 43, 44.7), | \(x_{14}, x_{31}, x_{36}, x_{52}, x_{55}, x_{62}\) |
\(29 \le z_{22} \le 43\), | (31.9, 33, 34.5), | ||||
\(30.3 \le z_{23} \le 44.7\) | (21, 22, 23.1) | ||||
3 | Quality | \(11.1 \le z_{31} \le 13\), | 0.8681 | (40, 41, 42.7), | \(x_{13}, x_{14}, x_{31}, x_{46}, x_{55}, x_{62}\) |
\(12 \le z_{32} \le 14\), | (41.9, 43, 44.6), | ||||
\(13.2 \le z_{33} \le 15.2\) | (11.1, 12, 13.2) | ||||
Aspiration level: (0.7, 0.8, 0.9) | |||||
1 | Cost | \(22.2 \le z_{11} \le 40\), | 0.8009 | (33.1, 34, 35.4), | \(x_{13}, x_{21}, x_{44}, x_{46}, x_{55}, x_{62}\) |
\(23 \le z_{12} \le 41\), | (47.7, 49, 50.9), | ||||
\(24 \le z_{13} \le 42.7\) | (16.9, 18, 19.2) | ||||
Shape parameter: (−2, −5, −1) | |||||
Aspiration level: (0.8, 0.85, 0.7) | |||||
1 | Cost | \(22.2 \le z_{11} \le 33.2\), | 0.8215 | (25.3, 26, 27.2), | \(x_{13}, x_{24}, x_{35}, x_{46}, x_{51}, x_{62}\) |
\(23 \le z_{12} \le 34\), | (61.6, 63, 64.9), | ||||
\(24 \le z_{13} \le 35.6\) | (18.9, 20, 21.2) | ||||
2 | Time | \(28 \le z_{21} \le 47.8\), | 0.7126 | (37, 38, 39.5), | \(x_{14}, x_{21}, x_{36}, x_{53}, x_{55}, x_{62}\) |
\(29 \le z_{22} \le 49\), | (39.8, 41, 42.7), | ||||
\(30.3 \le z_{23} \le 50.8\) | (22.9, 24, 25.2) | ||||
Shape parameter: (−1, −2, −5) | |||||
Aspiration level: (0.7, 0.8, 0.85) | |||||
1 | Quality | \(11.1 \le z_{31} \le 19,\) | 0.7972 | (33.2, 34, 35.6), | \(x_{13}, x_{14}, x_{46}, x_{51}, x_{46}, x_{62}\) |
\(12 \le z_{32} \le 20\), | (47.8, 49, 50.8), | ||||
\(13.2 \le z_{33} \le 21.2\) | (13, 14, 15.2) | ||||
Aspiration level: (0.8, 0.7, 0.75) | |||||
2 | Quality | \(11.1 \le z_{31} \le 21\), | 0.8191 | (32.1, 33, 34.5), | \(x_{11}, x_{13}, x_{24}, x_{46}, x_{55}, x_{62}\) |
\(12 \le z_{32} \le 22\), | (46.7, 48, 49.7), | ||||
\(13.2 \le z_{33} \le 23.2\) | (15, 16, 17.2) | ||||