Review of the main developments in the analytic hierarchy process

TLDR: This paper discusses problem modelling, pair-wise comparisons, judgement scales, derivation methods, consistency indices, incomplete matrix, synthesis of the weights, sensitivity analysis and group decisions that have appeared since its introduction.

Abstract: In this paper the authors review the developments of the analytic hierarchy process (AHP) since its inception. The focus of this paper is a neutral review on the methodological developments rather than reporting its applications that have appeared since its introduction. In particular, we discuss problem modelling, pair-wise comparisons, judgement scales, derivation methods, consistency indices, incomplete matrix, synthesis of the weights, sensitivity analysis and group decisions. All have been important areas of research in AHP.

Figures

Table 1: The 1 to 9 fundamental scale

Table 4: Four ways to combine preferences.

Figure 1: Example of a hierarchy (Akarte, Surendra, Ravi, & Rangaraj, 2001)

Figure 2: An example of four possible graphical sensitivity analyses in Expert Choice

Table 3: Random indices from (T. Saaty, 1977)

Table 2: Different scales for comparing two alternatives (for the comparison of A and B, c =

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Review of the main developments in the analytic hierarchy process" ?

In this paper the authors review the developments of the Analytic Hierarchy Process ( AHP ) since its inception. The focus of this paper is a neutral review on the methodological developments rather than reporting its applications that have appeared since its introduction. In particular, the authors discuss problem modelling, pair-wise comparisons, judgement scales, derivation methods, consistency indices, incomplete matrix, synthesis of the weights, sensitivity analysis and group decisions. 

Q2. What are the future works mentioned in the paper "Review of the main developments in the analytic hierarchy process" ?

The hierarchical modelling of the problem, the possibility to adopt verbal judgements and the verification of the consistency are its major assets. This phenomenon is still not fully resolved and maybe it will never be because the aggregation of preferences transposed from scales of different units is not easily interpretable and even questionable according to the French school ( Roy, 1996 ). The assumption of criteria independence ( no correlation ) may be sometimes a limitation of AHP ( and other MCDM methods ). The Analytic Network Process ( ANP ), a generalisation of AHP with feed-backs to adjust weights, may be a solution. 

Q3. What is the first method of arithmetical mean?

In the first method, the geometric mean of individual evaluations are used as elements in the pair-wise matrices and then priorities are computed. 

Q4. What is the only possible measurement for a weighted sum?

Saaty (1994) states that ratio scales are the only possible measurement if the authors want to be able to aggregate measurement, as in a weighted sum. 

Q5. How many comparisons can be deleted from a matrix?

In a Monte-Carlo simulation study, where comparisons are deleted from large matrices (rank 10, 15 and 20), it has been discovered that one can randomly delete as much as 50% of the comparisons without significantly reducing the results (Carmone, Kara, & Zanakis, 1997). 

Q6. What other software packages have been developed?

several other supporting software packages have been developed: Decision Lens, HIPRE 3+, RightChoiceDSS, Criterium, EasyMind, Questfox, ChoiceResults, AHPProject, 123AHP... not to mention that a template in Excel could also be easily generated. 

Q7. What scale is the often used in applications?

A > B; when A < B, the reciprocal values 1/c are used)Among all the proposed scales, the linear scale with the integers one to nine and their reciprocals has been used by far the most often in applications. 

Q8. Why does Barzilai claim that preference is not represented with ratio scales?

Barzilai (2005) claims that preferences cannot be represented with ratio scales, because in his opinion an absolute zero does not exists, as with temperature or electrical tension. 

Q9. What is the first step in the process of reversing the rank of the local priorities?

As the sum of the local priorities to unity changes with the introduction of a new alternative, the local priorities are also modified when normalised and therefore the global priorities may be reversed. 

Q10. Why do the authors believe rank reversals are more side-effects of the procedure than?

due to the absence of a causal effect demonstration, the authors believe that the occasional rank reversals are more side-effects of the procedure rather than credible results of the modelling procedure. 

Q11. What is the way to calculate the consistency index?

the consistency index can be calculated only based upon the entered comparisons (P. T. Harker, 1987b): the priority vector (pi, i = 1,2,...,n) is calculated (see section 2.6.1) which in turn is used to calculate estimates missing values (pi/pj, i = 1,2,...,n; j = 1,2,...,n). 

Q12. How do you calculate the mean of the rows?

Add the elements of the columns: (1.75; 7; 3.5)2. Normalize the columns:3. Calculate the mean of the rows: (a= 0.57; b=0.14; c=0.29).