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

Computing the exact ideal and nadir criterion values is a very ýimportant subject in ýmulti-ýobjective linear programming (MOLP) ýproblemsýý. In factý, ýthese values define the ideal and nadir points as lower and ýupper bounds on the nondominated pointsý. ýWhereas determining the ýideal point is an easy worký, ýbecause it is equivalent to optimize a ýconvex function (linear function) over a convex set which is a convex optimization problemý, ýbut the problem of computing ýthe nadir point in MOLP is equivalent to solving a nonconvex optimizationýproblem whose solving is very hard in the general caseý. ýýIn this paperý, ýa bi-level linear programming problem is presented for obtaining the nadirpoint in MOLP problems which can be used in general to optimize a ýlinear function on the nondominated setý, ýas wellý. Thený, ýas one of the solution methods of this problemý, ýaýmixed-integer linear programming problem is presented which obtains ýthe exact nadir values in one stageý.