This paper addresses a general multiobjective optimization problem. One of the most widely used methods of dealing with multiple conflicting objectives consists of constructing and optimizing a so-called achievement scalarizing function (ASF) which has an ability to produce any Pareto optimal or weakly/properly Pareto optimal solution. The ASF minimizes the distance from the reference point to the feasible region, if the reference point is unattainable, or maximizes the distance otherwise. The distance is defined by means of some specific kind of a metric introduced in the objective space. The reference point is usually specified by a decision maker and contains her/his aspirations about desirable objective values. The classical approach to constructing an ASF is based on using the Chebyshev metric L (a). Another possibility is to use an additive ASF based on a modified linear metric L (1). In this paper, we propose a parameterized version of an ASF. We introduce an integer parameter in order to control the degree of metric flexibility varying from L (1) to L (a). We prove that the parameterized ASF supports all the Pareto optimal solutions. Moreover, we specify conditions under which the Pareto optimality of each solution is guaranteed. An illustrative example for the case of three objectives and comparative analysis of parameterized ASFs with different values of the parameter are given. We show that the parameterized ASF provides the decision maker with flexible and advanced tools to detect Pareto optimal points, especially those whose detection with other ASFs is not straightforward since it may require changing essentially the reference point or weighting coefficients as well as some other extra computational efforts.