A dynamic clustering method to improve the coherency of an ANP Supermatrix

Abstract

When making decisions with the Analytic Network Process, coherency testing is an important step in the decision making process. Once an incoherent priority vector is identified it can either be costly or in some cases next to impossible to elicit new pairwise comparisons. Remarkably, there is useful information in the linking estimates that one may have already calculated and used in one of the approaches to measure the coherency of the Supermatrix. A dynamic clustering method is used to automatically identify a cluster of coherent linking estimates from which a new coherent priority vector can be calculated and used to replace the most incoherent priority vector. The decision maker can then accept or revise the proposed new and coherent priority vector. This process is repeated until the entire Supermatrix is coherent. This method can save decision makers valuable time and effort by using the information and relationships that already exist in a weighted Supermatrix that is sufficiently coherent. The method is initially motivated and demonstrated through a simple straightforward example. A group of conceptual charts and a figure provide a visual motivation and explanation of the method. A high level summary of the method is provided in a table before the method is presented in detail. Simulations demonstrate both the application and the robustness of the proposed method. Code is provided, as supplementary material, in the programming language R so the method can be easily applied by the decision maker.

Appendix

1.1 Calculating the linking estimates (LE)

Let the initial estimated Supermatrix be:

(12)

Every column of the Supermatrix should be converted to a new ratio by dividing the components of each vector by any common element within the priority vectors; the first element in each priority vector will be used herein. The resulting matrix will now have the form:

(13)

If the entries from one column in the lower hand side of the estimated Supermatrix in (13) are used as the links to convert the entries in the upper right hand side of the Supermatrix into quantities of a single unit, the converted upper right hand side will have the form:

(14)

With each entry in each column now represented in units of a particular, yet same, ratio as in (14) they can be aggregated and combined obtain a new estimate of S which we call a linking estimate. This new Supermatrix will be notated by $S^L_{C_{1.1}}$ since the criterion $C_{1.1}$ was used as the link. This estimate can be obtained by performing the following calculations:

(15)

where $T_{i.j,.}=\sum _{n=1}^3 T_{i.j,n}$ and $T_{..,n}=\sum _{i=1}^2 \sum _{j=1}^3 T_{i.j,n}$.

The same process can be repeated $n+m$ times where n is the number of alternatives and m is the number of criteria; a different criterion or alternative is chosen as the link each time, resulting in $n+m$ linking estimates. For detailed explanations about this concept please refer to Cooper and Yavuz (2016).