A New Approach to Solving Multiobjective Set-Valued Linear Fractional Optimization Problems Based on the Null Set Concept

Source abstract

We propose a hybrid solution approach for multiobjective set-valued linear fractional optimization problems, combining the null set concept with Dinkelbach's parametrization method. The approach proceeds in five computational stages. First, ideal values are computed for each fractional objective via scalar fractional programming. Second, the multiobjective set-valued fractional problem is converted into a multiobjective set-valued linear problem via Dinkelbach parametrization. Third, weighted-sum scalarization reduces it to a single-objective set-valued linear problem. Fourth, vectorization via the null set concept transforms this single-objective problem into a deterministic bi-objective linear problem solved by a single linear program. Fifth, an optimality test either certifies the current point or updates the parameter vector. A new biconditional theorem, in which the parameter is anchored at the point being tested, shows that Pareto optimality is preserved in both directions between the parametrized and the original problem, and yields a verifiable certificate: when the aggregate scalar objective vanishes at its minimizer, that minimizer is Pareto optimal. Under the additional assumption of a common vectorized denominator, the procedure is shown to terminate after finitely many iterations. We also introduce, for the first time in the set-valued fractional setting, notions of local and weak Pareto optimality and verify them on numerical examples. The effect of the weight vector on the recovered Pareto front is analyzed explicitly.