Group Aggregation Techniques for Analytic Hierarchy Process and Analytic Network Process: A Comparative Analysis

The trend towards more democratic organization structures, global networking and advanced technical possibilities often leads to strategic managerial decisions based on collective decision making and not on single decision makers. Organizations try to use the potential strengths of a collective decision making, assuming that a group of experts makes better or at least more objective decisions than individuals which underlie natural cognitive restrictions. Compared to single persons’ decision making, also termed as individual decision making (IDM), groups are providing the advantages of a broader amount of information, much more experience and alternatives, a better diversification of the individuals’ cognitive restrictions, less evaluation mistakes, and an increased acceptance of the solution (Sims 2002; Kreitner and Cassidy 2011). Participation in decision making processes may lead to a better utilization of knowledge as well as to a higher level of individual involvement and responsibility for finding an optimal solution of organizational problems (De Haas and Kleingeld 1999). However, at the same time group decision making also incorporates the danger of possible disadvantages like, for example, the domination of the group interaction by single group members with more assertiveness and vocal strength than other more quiet people (Kreitner and Cassidy 2011). Owing to this, the need for a systematic group decision support emerges.

In group decision and negotiation literature often the use of voting procedures as one of the fundamental principles to solve collective decision problems can be found. Voting procedures are based on the premise of decision problems to be solved by collectives within fixed procedures of democratic systems. They provide, for example, with majority-oriented voting rules the opportunity to determine ordinally scaled group preference orders. Areas of application include mass voting situations, like political elections, district or national referendums or parliaments with a huge amount of members, which have to be limited to the use of voting procedures already for practical reasons (less time, big group sizes, very heterogeneous group experiences and knowledge). All these decision contexts have in common that they demand from the decision makers only competence to judge in terms of a purely ordinal preference level. Additionally, the decision is implicitly compressed to a mono-criterion decision problem so that the decision makers have to mentally analyze, evaluate and assess all aspects relevant for their opinion simultaneously. Despite the overall favorability of such an approach, there are decision situations in which the functionality of voting procedures is not sufficient. One may think on situations in which the decision problem is so complex and extensive that its complete solution needs decomposition and new structuring as well as an intensive examination of its influencing criteria and dependencies and, furthermore, quantitative preference distances within the ranking of alternatives instead of ordinal preference orders. The solvation of such long-term oriented and highly aggregated strategic decision problems by (a restricted number of) members of a decision group with high individual expert knowledge implies the representation of complex individual information by adequate decision support methods as well as the necessity to apply group preference aggregation procedures. Although the well-known and often discussed voting procedures may lead to a group consensus even in such a strategic decision context, an interesting research gap can be identified in supporting strategic group decision making conducted by small expert groups with cardinally scaled and technically ambitious methods.

Moreover, such strategic decision problems are usually determined by multiple, differently dimensioned and partially conflicting goals. The complexity of strategic decision settings therefore requires not only a systematic, multi-criteria decision support but also the explicit consideration and systematization of information, experiences and preferences of multiple decision makers, whose participation may significantly improve the outcome of strategic decision environment.

The AHP (Saaty 1980; Saaty and Vargas 2001) and its generic form the analytic network process (ANP) (Saaty 1996; Saaty and Vargas 2013) are important discrete multiple criteria decision making (MCDM) methods gaining increasing importance in research and practice. Apart from their advantages in particular with regard to the ability of handling quantitative as well as qualitative information, efficient software support and the compensatory character of considering criteria, a further important benefit in comparison to other methods is their feasibility to consider decision processes adequate to reality, i.e. with multiple actors. So, AHP and ANP provide a wide range of enhancement options for multi-personal decision contexts. Although in cases of decision problems with high complexity the disadvantage of a very time-consuming application may arise, AHP and ANP offer in contrast to voting procedures the advantage of cardinally scaled priority results which measure the distances between the alternatives. Therefore, they offer a greater potential to solve the multi-personal decision problem than the above mentioned ordinal voting procedures. Referring to a MCDM-taxonomy, AHP and ANP can be positioned between ordinal voting procedures and approaches that require substitution rates and utility functions to represent the decision maker’s preferences. Insofar AHP and ANP differentiate from (more complex) utility theory based approaches as they use cardinal priorities to approximate individual preference structures instead of continuous utility functions. Utility function based approaches provide a more precise determination of optimal decision alternatives for the case if utility theory’s underlying conditions (e.g. substitutability) can be fulfilled. In situations when members of the decision group with highly individual expert knowledge shall collectively solve a strategic decision problem adequate to restrictive conditions of reality, the necessary explication of relevant utility function(s) often will not work. So, the decision settings mostly do not provide enough information for the construction of substitution rates between criteria and the evaluation of minimal variations of consequences. This will often fail in view of the decision makers‘ cognitive limits. For theoretical discussion of the important topic of cardinal utility in the context of collaborative group decisions see e.g. Keeney and Kirkwood (1975), Keeney (1976, 2009) as well as Dias and Sarabando (2012). Different to the multi-attribute-utility theory (MAUT)—as a typical representative of utility function based approaches—both AHP and ANP can be used equally for existing cardinal and ordinal information on attributes and generate cardinal rankings of the alternatives. By this, AHP/ANP suffice ambitious requirements, but are more robust for practical purposes than MAUT can be. Although this cardinality is an important scientific topic, it is not mentioned in most research contributions.

Nevertheless, keeping the identified research gap of strategic decision making with well-informed expert groups in mind, the application of AHP and ANP for group decision making (GDM) is not as unproblematic as its seems. In the field of MCDM research, many authors already recognize the immense importance of supporting group decision making. In recent years, numerous methods, especially for the well-known and often best suited AHP and ANP, have been developed. Faced with the variety of different approaches, however, the users both in science as well as in practice can lose orientation concerning the availability and appropriate selection of a group preference building method in the concerning decision making context. Therefore, a significant lack of systematization and assessment in scientific publications can be identified. Against this background, the development of a concrete view on existing approaches and of recommendations how to use them, can be concretized as identified research gap. As a vast number of aggregation approaches exist, the aim and new contribution of this research is on the one hand the presentation of an innovative and comprehensive literature review as well as on the other hand an evaluation of selected approaches of group preference building within the AHP and ANP regarding the demand criteria of the specific group decision problem to be solved.

The structure of this paper is as follows. Section 2 will point out the importance of AHP/ANP especially in the field of strategic GDM. Section 3 is focused on a literature review of the various approaches applicable for AHP/ANP group aggregation. The more detailed presentation and evaluation of selected approaches is the topic of Sect. 4 before the results of the evaluation and the comparison are summarized in Sect. 5. The paper ends with concluding remarks and future prospects in Sect. 6.

The AHP is a MCDM method for dealing with multiple goals and multiple criteria in complex decision settings (Saaty 1980; Dyer and Forman 1992; Saaty 2001b). The application of analytical methods, the decomposition and hierarchical structuring of the decision problem and a systematic, comprehensive framework form the procedure of the AHP method. The AHP is based on the three principles: decomposition, comparative judgments and synthesis of priorities (Saaty 1986; Dyer and Forman 1992). As a generic form of the AHP, the ANP provides further possibilities for structuring and analyzing complex strategic decision problems (Saaty and Vargas 2013). In contrast to the hierarchically structured AHP, the ANP facilitates explicit modeling of dependencies and feedback. Problems on a standard level of complexity should be solved by AHP, whereby an increasing connectivity is inducing the application of ANP. Both the AHP and ANP methodologies have been made a subject of various books and papers (see e.g. Dyer and Forman 1992; Forman and Gass 2001; Saaty 2001b, 2009; Saaty and Vargas 2013).

For underpinning the findings of past bibliometric investigations (see e.g. Vaidya and Kumar 2006; Ho 2008; Sipahi and Timor 2010; Hülle et al. 2013), we perform a comprehensive bibliometric study (see e.g. Van Raan 2003; White 2004; Jarneving 2005) for this paper in order to highlight the ongoing rising importance of AHP and ANP within literature. As we use the Business \(\hbox {Source}^{\circledR }\) Complete database via EBSCOhost, \(\hbox {SciVerse}^{\circledR }\) ScienceDirect and Thomson Reuters Web of \(\hbox {Science}^\mathrm{TM}\) to achieve a maximum scope of literature our analysis strongly exceeds all other bibliometric studies in this field. Our time horizon covers the last decade from 2004–2013 (database accesses for 2004–2011: July 16, 2012; for 2012: January 28, 2013 and for 2013: January 30, 2014). We consider all available publications within the databases. For obtaining significant results, the relevant journal contributions’ titles, key words and abstracts were searched by “analytic hierarchy process”/“analytical hierarchy process” and “analytic network process”/“analytical network process”. Owing to the fact that we use three databases, inconsistencies in the database’s coding as well as duplications had to be corrected manually. Figure 1 shows the synthesized and manually adjusted results.

With respect to the time distribution of the AHP and ANP publications there can be recorded a clear growth which furthermore can be interpreted as a rising importance of both approaches within research and practice (due to the large proportion of case studies). With respect to all MCDM methods the AHP shows the highest number of publications since the early 1990s (Wallenius et al. 2008; Sipahi and Timor 2010; Hülle et al. 2011). Potential fields of application range from selection, evaluation or allocation to cost-benefit-analysis and forecasting problems. Moreover, the AHP and ANP are beneficial tools for the support of strategic decision settings (Vaidya and Kumar 2006). Although the ongoing results may be transferable to the ANP the subsequent sections primarily focus on AHP due to some reasons. The AHP has the higher general awareness, its hierarchical structure is intuitively more understandable for inexperienced users and because of its simplicity it is more suitable for an illustration and evaluation of group aggregation techniques. In order to answer the identified research gap in terms of recommending an approach for group decision making in small expert groups a differentiated analysis of the AHP publications has been conducted subsequently. The articles published in the first and the last year of the examined time horizon has been further analyzed if an individual or a multi-personal decision making context (e.g. an explicit GDM, a survey of different experts or at least a group-based identification of decision parameters) is supposed. Figure 2 shows the results of this analysis in form of the total number of publications as well as the proportion of GDM in percent of the total number in 2004 and in 2013.

A comparison of the first and the last year examined in the conducted bibliometric analysis shows that both the total number and the relative proportion of GDM publications increased in the course of time. It shows that the relevance of GDM as well as the scientific interest in solutions and applications of decision support methods in a multi-personal environment is still existent. Against the background of an increasing number of AHP publications in general (see again Fig. 1) it is noticeable that despite of a growing range of different topics of AHP related research projects the quota of GDM articles remains not only constant over the time but even increases from 8.84 to 10.51 %. A total number of 80 publications which are embedded in a group decision making context in only one single year emphasize the relevance of an adequate multi-personal decision support and confirm the identified research gap. Furthermore, in the total of AHP publications an enrichment of facets of key words can be observed comparing 2004 with 2013. So the nominal increase of the GDM-proportion is—from a realistic point of view—even higher.

By the more detailed bibliometric analysis it could be shown that most strategic decision problems solved by AHP have in common that there is a need for including more than one decision maker in order to cope with the inherent complexity and to achieve a broader, holistic view on the decision environment. This poses the challenge of considering the individual expert judgments in the desired manner. As there is no generally accepted way for aggregating individual preferences/judgments to a consensus, the aim of this paper is to perform a comparative analysis of the different methods and “inside-approach” possibilities for aggregating the individual expert preferences to a GDM solution. “Inside-approach” possibilities describe the variants of aggregation procedures within a certain approach, for example the options of using the arithmetic or the geometric mean method inside the approach of aggregating individual judgments. In order to derive recommendations for varying demands within different decision settings, the subsequent section starts with a categorization of various group aggregation techniques. Furthermore, the most appropriate methods will be selected and evaluated with regard to certain criteria.

Although AHP and ANP were originally developed for an IDM context, both approaches have various methodological extensions for coping with a strategic multi-person GDM context. There are numerous group aggregation techniques that can be adapted to different group decision situations, taking into account individual as well as shared values (Dyer and Forman 1992; Aull-Hyde et al. 2006). Depending on the particular decision context, aggregation techniques differ strongly in complexity, available aid of mathematical auxiliary methods, consideration of risk or the adequate structure of the decision problem. Apart from striving for a consensus by discussion—when all group members basically have similar objectives—the basic consensus option is to intervene the decision making process at various stages and to aggregate either individual judgments or priorities (Dyer and Forman 1992). However, the structure of the group should be analyzed prior to the application of formal aggregation techniques. In some decision settings it is appropriate to waive the assumption of equivalent group members in favor of a differentiated, relative weighting of votes. Marked differences in knowledge, experience, management level or competence between the decision makers should be reflected in their respective influences on the overall ranking of the alternatives. In order to deal with inequalities with respect to the decision power of different group members, a supra decision maker could be determined who has the competence to assign weights to each group member in an exogenous manner. In cases where such a supra decision maker does not exist or does not seem to be acceptable for the participants, the group can try to allocate the voting power in an endogenous way. The endogenous decision power distribution could be arranged in such a way as the group is designing an additional hierarchy level to judge the relative power of each group member. Thereby, decision-relevant evaluation criteria could be decision makers’ personality traits such as perceived intelligence, experience, and power or management level. All participants could then be compared according to their relative influence with respect to underlying evaluation criteria (Saaty 1989, 2001a; Ramanathan and Ganesh 1994).

Having determined the weightings (\(w^{r}\) where \(r=1, \ldots , R; \mathop \sum \nolimits _{r=1}^R w^{r}=1\)) for all decision makers R, the subsequent step is the selection of an appropriate formal aggregation technique. As indicated above, the two fundamentally accepted aggregation procedures within AHP and ANP GDM are the aggregation of individual judgments (AIJ) and the aggregation of individual priorities (AIP).

In case the group structure is homogenous and decision makers are willing to act like one single individual, a synergistic AIJ is possible. Each decision maker conducts the pairwise comparisons by himself. Afterwards the (weighted) geometric mean method (WGMM) could be used to obtain the group judgment for each entry of the comparison matrices (Saaty 1989; Forman and Peniwati 1998). Hereby, the arithmetic mean should not be used which is due to the non-reciprocity (power conditions) of the collective pairwise comparison matrices (Aczél and Saaty 1983).

In case of a decision context attended by a conflict of interests, wherein the group members are individually acting with their own value systems, a consensus may be reached by using AIP. After each decision maker comes to his independent AHP or ANP based ranking, the resulting individual priorities are aggregated to a final group preference using either a (weighted) arithmetic (WAMM) or a (weighted) geometric mean method (Van den Honert and Lootsma 1996; Forman and Peniwati 1998).

Aside from AIJ and AIP, there is a multitude of further techniques to reach a group consensus (preference). In order to provide an adequate and structured literature review we use the taxonomy of six categories with comparable subject areas to classify the numerous publications in the field of AHP and ANP GDM. For achieving a transparent presentation, the identified methods and subjects areas are categorized by their characteristic features and specific application purposes. The result of this line of action is an innovative and comprehensive literature review which brings the new contribution to the AHP as well as to the group decision and negotiation literature of providing a clearly structured compendium of possible group aggregation approaches that simplifies the selection of appropriate methods and maybe even helps the reader to find an interesting topic for further personal research. The categorized literature review is presented in Table 1. The numbers in square brackets represent the related references in the “Appendix 1”.

Category I includes all publications which are inspired by the basic aggregation procedures and the consideration of direct information. The belonging publications deal with different AIJ and AIP approaches (see e.g. [18] or [20]) or present corresponding case studies (see e.g. [26] or [32]). If a real, practical group decision with high complexity should be solved, the user will certainly find an appreciable method in this first category which incorporates techniques with a broad applicability. In addition, this category offers the potential to support extensive case studies in a GDM context. So it probably shows the greatest relation to real decision problems.

Summarizing the above results, the innovative categorization of relevant publications helps to illustrate the wide range of possible AHP and ANP group decision support options. The variety of approaches and methods found while analyzing the database of the bibliometric analysis in Sect. 2 could be structured clearly and easily understandable. It provides for both users and researchers the opportunity to discover their own research gap or to get notes on any appropriate aggregation technique relevant for their specific interest. Despite of inherent hints that the first two categories may fit best to the small expert group based strategic decision context defined in Sect. 1, no recommendations which method is superior to others can be found yet. So, in terms of our research aim, the first part of answering the question which methods do exist has been finished. Nevertheless, no answer of the question which method should be used for solving highly complex decision problems in a small expert group could be found. The second part of evaluating how the identified and categorized methods can be used for a specific kind of decision problem is still open and therefore subject of the subsequent Sect. 4. As stated above in terms of a practical methods’ operationalization the approaches of the first two categories are in the particular focus for the subsequent representation and evaluation. Even though some of the approaches within the categories III to V seem to be theoretically very sound, practical limitations can be pointed out towards the fields of time effort and the elusive requirements of more sophisticated insights in complex mathematical techniques as well as additional OR-methods. Furthermore, a manageable number of evaluation alternatives has to be selected. Therefore, the AIJ and AIP methods (category I) as well as the loss function approach and the group aggregation on preferential differences (category II) are chosen for a more detailed representation and evaluation in order to identify an appropriate method to support GDM in the AHP.

To show particular differences with regard to certain decision environments AHP/ANP group aggregation techniques were evaluated with respect to relevant criteria. The evaluation was thereby supported by an AHP based case example and different test scenarios. Table 9 summarizes the identified results of the evaluation for the selected aggregation techniques and facilitates a direct comparison of their respective suitability.

In addition, Table 10 compares the resulting priorities and alternative rankings for each aggregation method. Obviously, it cannot be guaranteed that all methods indicate the same alternative as the best problem solution. While the procedure of AIJ and the LFA come to the conclusion that alternative \(A_2 \) should be chosen (case example, test scenario 1), the procedure of AIP and the Group AHP model recommend alternative \(A_1 \).

The aim of the performed evaluation is to support the selection of the most suitable aggregation method depending on the underlying decision environment.

However, the geometric variant (AIJ (WGMM)) shows an acceptable applicability for certain decision problems although only a small part of the available information is used (only individual judgments, see Table 5) and the fulfillment of both the Pareto Optimality axiom and the homogeneity condition is restricted. The individual decision makers should be willing to act as a synergistic unit and to give up the separate identification of their personal rankings in favor of the collective group preferences. Under such conditions a homogeneous group with fixed structures and common objectives, the AIJ (WGMM) provides the advantage of an improvement of the collective consistency level. Despite of possible inconsistent individual judgments, the geometric aggregation of collective pairwise comparison matrices generates a higher or even a complete consistency for the group judgments so that the quality of the decision can be increased.

The results of our evaluation imply a particular suitability of the AIP procedure. No other aggregation technique is able to fulfill a comparable number of evaluation criteria. AIP provides a wide range of application possibilities, although the amount of information that is used (only individual priorities, see Table 5) is quite small, too. Nevertheless, this priority based aggregation technique can be applied to deal with multi-personal decision making problems regardless of the group size, the group compilation or the decision setting. Furthermore, AIP shows great potential to support decisions with diverging or conflicting goals. Since AIP (WAMM) cannot guarantee the fulfillment of the power conditions, the AIP (WGMM) is even more suitable in the sense of a rational group decision support.

Regarding to a critical appraisal of this research, there can be pointed out that the fulfillment of consistency criteria was observed through a discrete examination and not by a continuous simulation study. Continuous simulation studies offer opportunities such as for the evaluation of reactions on varying consistency levels (see e.g. Aull-Hyde et al. 2006 for a simulation study on the consistency of geometric aggregated randomized pairwise comparison judgments). For this purpose of a broader, comparative evaluation with 7 objects, the use of scenarios instead seemed to be more fruitful compared to a restricted application of the aggregation approaches to a single matrix. The consideration of only one matrix can also be seen as unrealistic from practical settings’ point of view.

A further limitation towards practical implementation of nearly all approaches can be stated in the field of software support, as no current AHP/ANP Decision Support System provides any advanced feature for considering GDM. LFA and Group AHP model cannot be implemented by software. For AIJ and AIP several restrictions occur which depend on the underlying software product.

Complex and extensive strategic decision problems are often characterized by a great number of different influencing criteria and dependencies. So, the solution of such strategic problems needs not only the competences and experiences of a decision group with a (sufficient but not too large) number of persons having available highly individual expert knowledge, but also an adequate multi-criteria decision support method. Against the background of a multi-personal decision context with quantitative and qualitative criteria instead of often used ordinal voting procedures, special MCDM methods like AHP and ANP suit better. AHP and ANP are important MCDM methods for solving complex, strategic decision problems, which provide the possibility to generate cardinally scaled evaluations as well as a high inherent variety of group aggregation procedures. In contrast to other methods, especially to MAUT-methods which also additively aggregate, but need a continuous utility function, they must suffice less restrictive conditions, thus have a higher tolerance against practical imperfectness and therefore are to be considered as the more robust procedures on the background of the sketched group decision problem.

In suppose of methodical suggestions in the area of AHP and ANP their relevance for group decisions was investigated. Therefore, a comprehensive bibliometric analysis was performed in Sect. 2. With respect to the temporal development in literature, the results underpin the ongoing importance of these methods. Second, a more detailed analysis of the AHP database showed that the quota of publications related to a GDM context increased in the course of time, although lots of new research topics had been developed in the meantime. Due to the great variety of AHP and ANP group aggregation methods, the first aim of this paper was an identification and categorization of AHP- and ANP-based methods supporting GDM problems. So, GDM-relevant AHP- and ANP-literature was investigated and presented in a comprehensive literature review on various aggregation possibilities in Sect. 3 within which six categories were differentiated and explained. As a definite solution for highly complex decision problems in a small expert group could not be identified yet, different approaches of the first two categories were analyzed as the second aim of this relevant research.

To point out selected methods’ (AIJ (WAMM), AIJ (WGMM), AIP (WAMM), AIP (WGMM), LFA (WAMM), LFA (WGMM) and Group AHP model) particularities, the aggregation approaches were evaluated on the basis of their fulfillment of derived criteria in the fields of underlying decision context, consistency and social choice axioms in Sect. 4. Therefore, four test scenarios were introduced to ensure the consideration of the combinations’ possible variety.

The procedure of AIP offers at low time effort the highest fulfilment of relevant rationality axioms and the broadest range of applications. It is suitable for small and large groups as well as decision settings with common and conflicting goals. Under the restriction of a complex and time-consuming implementation the LFA provides an explicit consideration of the consistency of pairwise comparison judgments. If it can be assumed that the individual decision makers put aside their own interests in favor of a preferably small and homogenous group, the WGMM variant of AIJ may be also recommended. Its arithmetic variant, however, cannot be used. Although the Group AHP model integrates a high degree of available information its applicability is rather limited to small groups and simply structured decision problems. So, the AIP remains as the most recommendable aggregation technique within the AHP and ANP to support highly complex decision problems in a small expert group.