An interval number programming approach for bilevel linear programming problem

A novel approach based on preference-based index for interval bilevel linear programming problem

“…This example extracted from the literature [ 22 ] is an interval bilevel linear programming problem with only one interval inequality constraint: …”

“…For this example, Abass [ 22 ] gives the optimal solution as with the possibility degree level by adopting the midpoints and half-widths of the interval coefficients to formulate both interval objective functions and applying the concept of the possibility degree of interval number to treat interval inequality constraints. As far as the final result is concerned, the result obtained by our approach is the same as that obtained in [ 22 ].…”

“…For this example, Abass [ 22 ] gives the optimal solution as with the possibility degree level by adopting the midpoints and half-widths of the interval coefficients to formulate both interval objective functions and applying the concept of the possibility degree of interval number to treat interval inequality constraints. As far as the final result is concerned, the result obtained by our approach is the same as that obtained in [ 22 ]. From the viewpoint of equivalent transformation of interval inequality constraint, our approach can ensure that the formulated stochastic constraint at the obtained optimal solution holds with probability of at least 95% based on chance-constrained programming, in other words, this constraint is violated with probability of at most 5%.…”

“…From the viewpoint of equivalent transformation of interval inequality constraint, our approach can ensure that the formulated stochastic constraint at the obtained optimal solution holds with probability of at least 95% based on chance-constrained programming, in other words, this constraint is violated with probability of at most 5%. While equivalent conversion based on the possibility degree of interval number in [ 22 ] holds at a small possibility degree level . As we know, a smaller possibility degree means a lower reliability of the interval constraint.…”

“…Some studies as regards the interval bilevel linear programming problem exist in the literature. Among them, Abass [ 22 ] applied the possibility degree of interval number to reduce interval inequality constraints into deterministic inequality forms and employed the midpoint and radius of interval number to convert interval objective functions of the upper and lower levels into both deterministic objective functions. Then the interval bilevel linear programming problem with interval coefficients in both objective functions and constraints was transformed into a deterministic bilevel optimization problem.…”