In this work, we propose a new scheme for scalarization in set optimization studied with the Kuroiwa set appoach. First, we define an abstract scalarizing function possessing properties such as global Lipschizity, sublinearity, cone monotonicity, cone representation property, cone interior representation property and uniform positivity. Next, we use this function to define the so called signed Hausdorff-type half-distances and Hausdorff-type distances. As first applications, we obtain characterizations of Kuroiwa's set order relations and some optimal solutions of a set optimization problem. This scheme provides a unified approach to scalarization involving different scalarizing functions such as the Gerstewizt (Tammer) function, the Hiriart-Urruty signed distance and the function proposed by Kasimbeyli.