## Theorethical Background

The ADAM method represents entirely new multi-criteria decision-making (MCDM) method. The process of ranking alternatives in this method is based on determining the volumes (aggregated measurements) of complex polyhedra that are defined by points (vertices) in a three-dimensional coordinate system. Each of the points belongs to one of three types: coordinate origin (O), reference points (R), and weighted reference points (P). The coordinate origin is a point with coordinates (0,0,0). Reference points are points with coordinates (x,y,0) that define the value of the alternative according to some criterion as the axial distance of the point from the coordinate origin in the x–y plane. The weighted reference points have coordinates (x,y,z), where the coordinate z is used to obtain the axial distance of the weighted reference point from the x–y plane. These distances correspond to the weights of the criteria. A simple example of a complex polyhedron obtained for the problem of evaluating an alternative in relation to the three criteria is shown in the following figure.

An example of a complex polyhedron for evaluating an alternative against three criteria.

The complex polyhedron shown in the figure is defined by seven points, of which O indicates the coordinate origin; points R, R2, and R3 indicate the reference points and points P1, P2 and P3 indicate weighted reference points. The coordinates of the reference points R1, R2, and R3 are obtained on the basis of a vector that defines the value of the alternative according to some criterion. A vector is defined by course, direction, and magnitude. The course of the vector that defines the value of the alternative is defined by the angle α in relation to the x axis. The vector is always directed away from the coordinate origin. The magnitude of the vector is defined by its length, which corresponds to the value of the alternative according to that criterion.

In the example in the figure, the coordinates of the reference point Q for the first criterion are obtained by defining the vector (R1), whose course is defined by an angle of 0° in relation to the x axis, starting from the coordinate origin and ending at a distance of 3 from the coordinate origin. Accordingly, it is clear that point R1 has coordinates (3,0,0). In a similar way, the coordinates of the reference point R2 for the second criterion are obtained through the vector (R2), whose course is defined by an angle of 45° with respect to the x axis, starting from the coordinate origin and ending at a distance of 4 from the coordinate origin. Accordingly, it is clear that point R2 has coordinates (2.82,2.82,0). The coordinates of the reference point R3 for the third criterion are obtained through the vector (R3), whose course is defined by an angle of 90° in relation to the x axis, starting from the coordinate origin and ending at a distance of 2 from the coordinate origin. Accordingly, it is clear that point R3 has coordinates (0,2,0).

The coordinates of the weighted reference points P1, P2, and P3 are obtained by adding a third dimension to the projections of the points R1, R2 and R3 from the x–y plane; that is, the coordinate z, instead of a value of 0, takes the value of the weight of the observed criterion. Accordingly, if the first criterion has a weight of 1, the coordinates of the point P1 are (3,0,1); if the second criterion has a weight of 3, the coordinates of the point P2 are (2.82,2.82,3); and if the third criterion has a weight of 2, the coordinates of point P3 are (0,2,2).

By defining all the points that define a complex polyhedron, it is possible to calculate its volume as the sum of the volumes of k polyhedra (k=1,…,n-1), which are, in this case, irregular pyramids. Each polyhedron consists of sides formed by a coordinate origin with reference and weighted reference points of two consecutive criteria. The final order of alternatives is obtained according to the decreasing values of the obtained volumes of complex polyhedra.

The main advantages of the ADAM method in comparison with the aforementioned ones are that it is simple, easily understandable, user friendly, insensitive to the increase in the number of criteria, very intuitive, and has very low risk of rank reversal. The simplicity of the method is reflected in the use of simple calculations of the volumes of geometric bodies, i.e., polyhedra. Although these polyhedra are called complex, the calculation of their volume is actually very simple and requires a basic knowledge of geometry. Accordingly, the method is very simple to apply, and the calculation of final values for alternative ranking does not become more complex with an increase in the number of criteria. As the method is based on the volumes of geometric bodies that can be easily represented graphically, it is very easy to interpret the results and reach a conclusion about the final ranking of the alternatives. In most cases, this can be performed even visually, which makes the method very intuitive. The tests of the method proved that the obtained solutions are very stable and that changes in the criteria weights only insignificantly change the results, i.e., the risk of a change in the ranking of alternatives is very low.