350 – An attempt to explain Rank Reversal (RR) and reasonable inferences for its manifestations

Nolberto Munier

Given a set of different alternatives or products subject to a set of criteria or conditions, it is of great interest to determine which is the best alternative. This is a very common scenario in multiples fields, for instance, the selection of the best site, among several (or alternatives), to install an industry.

This is normally a difficult problem, however, a powerful procedure, Multi Criteria Decision Making (MCDM), is a technique that allows finding a solution, if not optimal, at least convenient for everybody. It not only determines the best site, but also ordains the other sites in decreasing order of importance, i.e., generates a ranking.

Sometimes, after a ranking has been obtained, there is necessity to add new alternatives, or delete some existent, and this may produce an alteration of the original ranking. If the original ranking has ordered four alternatives, for instance, A4 >A1 > A2 > A3, it could happen that if we add alternative A5, the original ranking transforms in A2 > A4 > A5 > A1 > A3. Observe that now A2 precedes A4, and thus, reversing the original ranking.

A5 can be in any place in the new ranking, but the primitive sequence must be respected. For instance, if the result is A4 > A5 > A1 > A2 > A3, observe that the sequence A4, A1, A2, A3 has not been altered. This is what should be, but in many cases it does not happen.

This is called Rank Reversal (RR) and it may appear in most MCDM methods. On course, this is an undesirable problem, because in theory, the addition or deletion of an alternative, or site, shouldn’t produce changes in the original ordering.

Belton and Gear (1983) discovered Rank Reversal (RR) in the Analytic Hierarchy Method (AHP). There are different theories and hypothesis trying to explain why RR occurs, mostly related to AHP, however, RR may happen in different methods other than AHP. There is also one intriguing question, and it is why some MCDM assert that they are immune to

Are they? Most probable not.

For this author, it appears that RR depends on the way alternatives are selected, and perceives that it occurs when an alternative is chosen considering other alternatives at the same time. Nevertheless, considering one MCDM method, it is weird that RR seems to appear at random in a problem, and absent in another.

That is, why a method solving a problem is affected by RR, and in other occasions, the same method, treating a different problem, is not affected?

This author hypothesizes that RR, whatever the method, is produced when the adding or deletion of one alternative involves also other alternatives, when it should be an individual action, that is, subject to their own values and characteristics.

As a trivial example, and using only reasoning and common sense: A person looking to buy a house, has a list of five houses that he already inspected, and made a mental ranking of them. Now he sees another house an adds it to his list. Why the addition of a house should alter his previous selection? If house C was in his former list better than house A, adding a new one shouldn’t change the original precedence.

The new house will modify the ranking in its extension (6 houses instead of 5), but not its precedencies. Thus, the new house could be put first, second, intercalated, or be the last in the ranking, but why the original precedencies should change?

The ample diffusion of the RR phenomenon, occurring in methods with different structures suggests that something common affects them all, and the purpose of this communication is to find out what it is.

Therefore, the objective of this paper is:

a) Find the reason for RR occurrence, and

b) Determine why it appears to be at random, even for a same method.

While different explanations have been given assuming that the culprit is normalization, or the inconsistency of the preferences, or the use of certain scales, etc., this paper assumes a different approach that apparently, according to this author knowledge, has not been considered before.

This approach is that RR may occur due to something that people apparently do not consider. It is the fact that when adding or deleting an alternative there is a switch into a different dimensional space.

For instance, from 2D to 3D when adding, and from 3D to 2D when deleting. And this is fact, not a hypothesis.

As a very coarse and simplified explanation. A space is defined by the number of its bases. i.e., a 2D space has as a base two dimensions, x and y, the coordinate axes.

A 3D space is defined with three axes x, y and z.

A 4D space is defined by four axes x, y, z and w, but we cannot see the fourth space, because we live in a 3D space

A 15D space is defined by fifteen axes, and so forth.

Consequentially, the number of alternatives defines the dimension of a problem.

This is extremely important in RR.

Let’s see

We can consider a simple problem with two alternatives (x1 and x2), and three criteria (C1, C2, C3). Alternatives and criteria have the same structure; both are mathematical equations.

Figure 1 displays the problem graphically. Observe the straight lines for criteria C1, C2 and C3, (in blue) in a two-dimensional space (alternatives x1 and x2, in black). It can also be seen. the objective function Z (in green), also in 2D. All are linear equations.

Considering the maximising and minimizing actions, that is, benefits or costs, some of the three criteria may form a geometrical figure (shown in thick red lines) with four vertices. Criteria lines for C1and C3 intersect in one of them (vertex identified as a solid black circle), this happens ONLY if the problem is feasible, for, if it is unfeasible, there is not a polygon.

📷 📷

📷📷📷 x2

Z

📷📷📷 Z’ Sense of Z displacement

📷📷📷📷📷📷📷📷📷 0.18

📷 0.10 C2

📷📷📷 0 0.15 C1 x1

0.20 C3

Figure 1 The problem initial matrix in graphical form

Assuming in this example that the three criteria call for maximization (although there can be any mix of maximization and minimization criteria, but here it is considered that the three criteria call for benefits, in order to facilitate the figure). Consequently, the area beneath each line and the coordinate axis x1and x2, represent all possible values that can take that criterion.

Notice than only C1 and C3 intersect, and in so doing, forming a polygon, that contains all the feasible solutions of the problem. The infinite points in the polygon satisfy simultaneously both criteria. Observe that C2 does not intersect with another line, and thus, it does not participate in the solution of the problem. Criteria C1and C3, that define the different solutions are called ‘Binding criteria’.

As seen, the sides of the polygon are formed by parts of the criteria lines, and at their intersections create a series of vertices.

There are three vertices other than the origin, and the optimal solution must be in one of them (Linear Programming Fundamental Theorem). Each alternative has a factor, for instance 3.2 x1 and 1.78 x2, and the sum of the product of these factors by the still unknown scores of the alternatives, constitutes the objective to maximize, minimize or equalise. This expression is the objective function (Z). It is also a line in the x1-x2 solution space materialized in a 2D coordinate system.

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This figure is known as a polygon in 2D, and it has the property that, if the problem is feasible, all infinite feasible solutions are contained in it. However, we are instered in determining which of those is the best.

If Z (3.2x1+ 1.78x2), the objective function, must be maximized, is displaced parallel to itself (dashed green line), until it tangents the polygon in one of its vertices (identified is a black solid dot), and this is the optimal solution, that gives the corresponding values of x1 = 0.20 and x2 = 0.10 (not at scale).

Therefore, the best alternative is x1 and the ranking is x1 > x2

Z = 3.2*0.20+1.78*0.10= 0.818, and this solution is optimal, implying that there is not a better solution (Pareto principle).

As an experiment, we can displace criterion C3 over criterion C1; in so doing, we will get an intersection (solid black rectangle) to the left, producing an increase in x2 score (0.18), as well as a decrease in x1score (0.15). Now, the ranking would be x2 > x1, that is, we reversed the ranking at purpose.

With these elements, the first aim of our problem, i.e., finding what produces RR may be addressed.

This author considers that changing dimensional spaces is the key to understanding rank reversal. When we add an alternative or delete an existing one, we are changingthe dimensional space of our problem, and in a two dimensions problem like the illustrated, we are switching from a 2D polygon, to a 3D polyhedron, and this is the reason for RR. This can be seen in Figure 2.

📷

📷 x2

Z plane due to adding x3

📷📷📷 Sense of Z displacement

📷📷

📷

📷📷📷📷📷📷📷📷📷 Tangency vertex x1

Polyhedron

📷 Plane Z’

x3

Figure 2 The same problem but adding alternative x3.

Observe that now we no longer have a Z line, but a Z plane, as well as no longer a polygon but a polyhedron. Since the objective calls for maximization, we can displace this plane parallel to itself until its tangents a vertex of the polyhedron, that contains all the feasible solutions of the problem, and being the optimal in one of the vertices. We can see that the Z’ plane tangents the polyhedron in one of the vertices, identified with a large solid black circle.

Why the plane tangents this special vertex? Because the slope of the plane has changed little regarding the Z line in 2D. Since the Z’ plane is pivoting on this vertex, observe that now the original ranking we had in 2D holds, although it does not have 2 scores but 3. Since its projection is now say 0.27, the new ranking is x3> x1 > x2, and thus, there is not RR, since the original precedence holds.

It can be appreciated that the selection of the tangency vertex may depend on the slope of the Z’ plane. Suppose that the Z plane has considerably changed its slope, due to the addition of x3. In that case, it can tangent the vertex that is on the x2 coordinate, and meaning that x1 is now 0.

Therefore, the ranking could be x3 > x2 > x1. In other words, there is RR, since the original precedence has been altered.

It could also be that the Z slope has drastically changed in the opposite sense, and now the plane is close to be vertical. In this case, the tangency would be in the vertex formed at x1. What does of mean? That x2 is now 0.

In a complicated problem with many criteria and alternatives the polyhedron could be a very complicated figure, called a polytope, which naturally, we cannot see, and with many vertices. In this case, deleting an alternative can make that the hyperplane (more than 3 alternatives), to tangent any other vertex, and then producing, again RR.

Observe that the objective plane, similar to the objective line in 2D can oscillate or pivot around a vertex, that is, changing its slope, but in not enough degree to alter the ranking, and then, no producing RR.

With this reasoning, it is believed that:

a) The reason for RR is explained

b) The reason by which, for the same problem and the same MCDM, there could be or not RR, has also been demonstrated.

In the first example in 2D, it was seen that we could artificially create RR by displacing one of the lines on another, and we saw that the optimal solution was given at the tangency of the objective function line Z with a vertex of the polygon. It can already be seen that an alteration of the slope of Z can produce RR, but we were doing that, keeping the same dimensional space.

It can be observed that this mechanism is replicated when we add or delete alternatives, with the only difference that we no longer have lines or polygons, but planes, or polyhedrons, and hyperplanes.

But we are not doing this at purpose; it depends on how the addition or deletion of a new alternative modifies the slope of the line or the hyper plane. This is, for this author the reason for RR.

Comments as well as criticism will be most welcome.