A bi-level linear programming problem for computing the nadir point in MOLP

Computing the exact ideal and nadir criterion values is a very ‎important subject in ‎multi-‎objective linear programming (MOLP) ‎problems‎‎. In fact‎, ‎these values define the ideal and nadir points as lower and ‎upper bounds on the nondominated points‎. ‎Whereas determining the ‎ideal point is an easy work‎, ‎because it is equivalent to optimize a ‎convex function (linear function) over a convex set which is a convex optimization problem‎, ‎but the problem of computing ‎the nadir point in MOLP is equivalent to solving a nonconvex optimization‎problem whose solving is very hard in the general case‎. ‎‎In this paper‎, ‎a bi-level linear programming problem is presented for obtaining the nadirpoint in MOLP problems which can be used in general to optimize a ‎linear function on the nondominated set‎, ‎as well‎. Then‎, ‎as one of the solution methods of this problem‎, ‎a‎mixed-integer linear programming problem is presented which obtains ‎the exact nadir values in one stage‎.