1 Introduction
2 Preliminaries
3 Interval bilevel linear programming problem
4 Solution methodology
4.1 Conversion of the interval constraint region into its crisp one
4.2 Preference-based deterministic bilevel programming problem
4.3 Solution approach based on estimation of distribution algorithm
5 Numerical examples and discussion
5.1 Numerical examples
5.2 Results and discussion
Interval
|
\(\boldsymbol {N(\mu,\sigma^{2})}\)
|
Interval
|
\(\boldsymbol {N(\mu,\sigma ^{2})}\)
|
---|---|---|---|
[3,5] |
N(4,0.33332) | [4,6] |
N(5,0.33332) |
Interval
|
\(\boldsymbol {N(\mu,\sigma^{2})}\)
|
Interval
|
\(\boldsymbol {N(\mu,\sigma ^{2})}\)
|
Interval
|
\(\boldsymbol {N(\mu,\sigma^{2})}\)
|
---|---|---|---|---|---|
\([\frac{34}{35},1]\)
|
N(0.9857,0.00482) |
\([\frac{17}{10},2]\)
|
N(1.85,0.052) |
\([10,\frac{51}{5}]\)
|
N(10.1,0.03332) |
\([-\frac{7}{6},-1]\)
|
N(−1.0833,0.02782) |
\([\frac{7}{5},2]\)
|
N(1.7,0.12) |
\([-6,-\frac{161}{30}]\)
|
N(−5.6833,0.10562) |
\([-\frac{5}{2},-2]\)
|
N(−2.25,0.08332) |
\([\frac{1}{2},1]\)
|
N(0.75,0.08332) | [−21,−20] |
N(−20.5,0.16672) |
\([-\frac{63}{40},-1]\)
|
N(−1.2875,0.09582) |
\([-\frac{21}{10},-2]\)
|
N(−2.05,0.01672) |
\([-38,-\frac{1407}{40}]\)
|
N(−36.5875,0.47082) |
\([\frac{7}{15},1]\)
|
N(0.7333,0.08892) |
\([-\frac{21}{10},-2]\)
|
N(−2.05,0.01672) |
\([-18,-\frac{84}{5}]\)
|
N(−17.4,0.22) |
Interval
|
\(\boldsymbol {N(\mu,\sigma^{2})}\)
|
Interval
|
\(\boldsymbol {N(\mu,\sigma ^{2})}\)
|
Interval
|
\(\boldsymbol {N(\mu,\sigma^{2})}\)
|
---|---|---|---|---|---|
[0.5,1] |
N(0.75,0.08332) | [1.9,2] |
N(1.95,0.01672) | [10,10.5] |
N(10.25,0.08332) |
[−2,−1] |
N(−1.5,0.16672) | [1,2] |
N(1.5,0.16672) | [−6,−5] |
N(−5.5,0.16672) |
[−3,−2] |
N(−2.5,0.16672) | [0.5,1] |
N(0.75,0.08332) | [−21,20] |
N(−20.5,0.16672) |
[−2,−1] |
N(−1.5,0.16672) | [−3,−2] |
N(−2.5,0.16672) | [−38,−37] |
N(−37.5,0.16672) |
[0.5,1] |
N(0.75,0.08332) | [−3,−2] |
N(−2.5,0.16672) | [−18,−17] |
N(−17.5,0.16672) |
γ
|
\(\boldsymbol {(x^{*},y^{*})}\)
|
\(\boldsymbol {F^{I}}\)
|
\(\boldsymbol {\delta^{F}}\)
|
---|---|---|---|
0.3 | [12.4317,9.2434] | [−9.2434,−4.6217] | 1.4865 |
0.5 | [12.4317,9.2434] | [−9.2434,−4.6217] | 1.6082 |
0.7 | [12.4317,9.2434] | [−9.2434,−4.6217] | 1.7298 |
0.9 | [9.1441,11.0] | [−11,−5.5] | 2.0316 |