Bi-objective robust optimisation

Robust solutions to multi-objective linear programs with uncertain data

“…Obviously, the concept of highly robust solution is quite exigent while the concept of flimsily robust solution is very permissive. Other concepts of robust solutions are compared with the previous ones in Ide and Schöbel (2013) and Kuhn et al (2013) while Pando et al (2013) provide conditions guaranteeing the maintaining of solutions through the elimination or aggregation of objectives. When the uncertainty affects both the constraints and the objective functions, the concepts of robust solutions are the same as in the latter paragraph except that, in (RP) each uncertain constraint g j (x) ≤ 0 is replaced by sup g j (x) ≤ 0 (the supremum taken over all possible scenarios).…”

“…In this paper, we provide some answers to the above questions for the uncertain multi-objective linear programming problem (P) in the face of data uncertainty by focusing on two choices of the robust optimal solutions: the first one is called a minmax robust efficient solution or simply robust efficient solution following the approach widely used in robust scalar optimization problem (see also Ehrgott, Ide, and Schöbel (2014); Kuroiwa and Lee (2012) for recent development), and corresponds to an efficient solution to a deterministic worst-case (minmax) multi-objective optimization problem; the second one is called highly robust efficient solution as in Ide and Schöbel (2013) and Kuhn, Raith, Schmidt, and Schöbel (2013) (see also Sitarz, 2008;Pando et al, 2013, Section 4), and consists of the preservation of the efficiency for all (c 1 , . .…”

Multi-objective minmax robust combinatorial optimization with cardinality-constrained uncertainty

“…Cardinality constrained uncertainty has been extended to multi-objective optimization in [DKW12] (only for uncertain constraints) and [HNS13] (for uncertain objective functions and constraints). To the best of our knowledge, only Kuhn et al have developed a solution algorithm for multi-objective uncertain combinatorial optimization problems [KRSS16]. They consider problems with two objectives, of which only one is uncertain, with discrete and polyhedral uncertainty sets.…”

On computing highly robust efficient solutions

“…Independently of the interval multiobjective programming studies, Kuhn et al () and Dranichak and Wiecek () have recently provided methods for computing HRE solutions to specific classes of UMO(L)Ps. Kuhn et al () present a brute‐force procedure to compute subsets of HRE solutions to an uncertain biobjective problem in which one objective is deterministic and the other uncertain. Otherwise, Dranichak and Wiecek () provide a robust counterpart, which is a deterministic MOLP whose efficient solutions may be easily obtained using existing methods (refer to Wiecek et al, ) and are HRE solutions to MOLP( U ), for a class of problems satisfying a special property.…”

“…The focus of this paper is uncertain multiobjective linear programs (UMOLPs) in which only the objective coefficients are uncertain. As in, e.g., Ehrgott et al (2014), Ide and Schöbel (2015), Kuhn et al (2016), and Dranichak and Wiecek (2018), the uncertainty may be restricted to the objective coefficients, and so the feasible set is taken to be deterministic.…”