A Spreadsheet Module for Consistent Consensus Building in AHP-Group Decision Making

Group Decision and Negotiation 14: 89–108, 2005.DOI: 10.1007/s10726-005-2407-8 C© Springer 2005

A Spreadsheet Module for Consistent ConsensusBuilding in AHP-Group Decision Making∗

JOSE MARIA MORENO-JIMENEZ, JUAN AGUARON JOVEN, AGUSTIN RALUY PIRLAAND ALBERTO TURON LANUZAGrupo Decision Multicriterio Zaragoza, Facultad de Economicas, Universidad de Zaragoza, Spain(E-mail: [email protected])

Abstract

This paper presents a spreadsheet module developed with Microsoft Excel and Visual Basic to search for consensuswhen dealing with AHP in group decision making. On the basis of the theoretical results included in the paperMoreno-Jimenez et al. (2003), this spreadsheet module allows us to construct, in a local context (one criterion), theConsistent Consensus Matrix (CCM) corresponding to the core of consistency associated with the actors involvedin the decision making process. By means of simulation techniques, this module provides, for each of the differentvalues of the inconsistency thresholds considered, the preference structures derived from the interval incompletematrix (CCM) that represents the group’s core of consistency. This information will be used to establish consensuspaths between the actors involved in the resolution process, one of the most important goals in distributed andparticipatory decision making.

Key words: group decision making, negotiation, AHP, consensus building, core of consistency, excel

1. Introduction

The holistic vision of reality that characterizes new scientific approaches followed in de-cision making (Moreno-Jimenez et al. 1999, 2001; Moreno-Jimenez 2003), together withthe new communication technologies, have motivated a significant amount of attention andinterest in the development of new decisional tools (analytic and informatic) which allowfor the participation in the decisional process of all the actors involved in the resolutionprocess. The complexity of the problems under study, especially those arising in the socialscientific field, where the human factor is essential for the resolution process and the in-creasing professional specialization of citizens requires the joint consideration in decisionmaking process of multiple actors that are usually spatially distributed.

In this sense, it is necessary to develop new informatic tools that, based on the use of thenetwork as communication support and the new decisional paradigms as methodologicalsupport, assist in decision making with multiple scenarios, actors and criteria.

∗This paper has received partial financial support from the Research Project of the University of Zaragoza(2002): “Herramientas decisionales para el constructivismo cognitivo” (ref. 228-056). A previous version of thispaper has been presented at the Group Decision and Negotiation session of the EURO-INFORMS Congress inIstanbul, 2003.

90 MORENO-JIMENEZ ET AL.

In what follows, and assuming that the analytic hierarchy process (AHP) is the multi-criteria technique used in the scientific resolution of the problem, we present the differentcomponents of a new informatic tool. This proposal, based on the theoretical results includedin Moreno-Jimenez et al. (2003), allows us to provide patterns of behavioural, trends anddecision opportunities derived from the incomplete pairwise comparison matrix correspond-ing to the “group’s core of consistency”, and, thereafter, to analyse their evolution whenvarying the inconsistency level considered for the study.

These patterns of behavioural, or preference structures associated with the ranking of thecompared alternatives (Moreno-Jimenez and Vargas 1993), will be taken as the output of thegroup decisional process when this has been agreed by the actors involved in the decisionmaking process. Alternatively, it could be considered as a previous step for consensussearching in the negotiation processes.

In the first case, the decision would be taken without a personal participation of theactors (automatic decision making), except at the judgment emission moment. However, inthe second case, the knowledge about the problem derived in the resolution process will beused as a starting point in the construction of consensus paths between the actors involvedin the decision making process. In this case, the interaction of the different actors (personalor consensual decision making) is obviously allowed.

The rest of the paper has been structured as follows. Section 2 includes the background ofthe work, particularly the theoretical results that, based on the concept of consistency, justifythe procedure followed. Section 3 presents the Excel module that provides, for the differentlevels of inconsistency, the preference structures derived from the interval judgment matrixcorresponding to the actor’s core of consistency. For a better understanding of this module,in Section 4 we apply it, first, to a case in a local context, that is to say, with one criterion,and, secondly, to a more general case, with two criteria. In both cases, the context is one oftwo decision makers and four alternatives. Finally, Section 5 highlights the most outstandingconclusions of the work.

2. Consistent Consensus Building in AHP-GDM

Decision making is one of the essential activities of humankind, and gives us an ideaof its degree of evolution, wisdom and freedom. In the past, scientific resolution hasbeen oriented towards the search for a unique and universal truth, by means of a rationaltreatment of the objective. Today, the resolution of high complexity problems, understoodas the sequence of actions that allows us to go from an initial situation to a desiredone, is oriented towards the search for the relevant knowledge of the decisional process,through an objective treatment of the rational and the emotional (Moreno-Jimenez2003).

This knowledge, understood in the sense of artificial intelligence as the interpretation ofthe information in a specific domain, requires the incorporation in the decisional process ofall the actor’s visions of reality, something that the traditional meaning of science did notallow, due to the value neutrality supposed by it.

A SPREADSHEET MODULE FOR CONSISTENT CONSENSUS BUILDING 91

Jointly with the incorporation of the subjective and intangible into the decision makingprocesses, one of the requirements suggested for the new scientific approaches followed indecision making is that these approaches must exhibit good behaviour for individual andgroup decision making (Saaty 1996; Moreno-Jimenez 2002).

In this sense, the AHP is a multicriteria technique, or more generally, a philosophyproposed by Saaty (Saaty 1980), that allows for the resolution of problems with multipleactors and criteria (tangible and intangible) by means of the construction of a ratio scalecorresponding to the priorities of alternatives.

The initial AHP, or Conventional-AHP, consists of three stages: (i) modelization; (2)valuation and (3) priorization and synthesis. The first deals with the construction of thehierarchy that represents the problem; the second incorporates the judgments that reflectthe preferences of the actors involved in the resolution process; finally, the third stageprovides the priorities of the compared alternatives.

Saaty (1980) proposed the use of the eigenvector method to obtain the local priorities,or relative priorities of the elements that hang from a common node. Similarly, he argued infavour of the hierarchical composition principle to obtain the global priorities, or prioritiesof the elements of a hierarchy with respect to the global goal or mission. Finally, he re-comended the application of an additive synthesis procedure to derive the final or totalpriorities of the alternatives.

One of the more outstanding aspects of AHP, that differentiates it from othermulticriteria techniques, is that it measures the inconsistency of the actors when elic-iting the judgments in a way that is formal, elegant and intrinsic to the mathematicalprocedure.

Defining the consistency as the cardinal transitivity in the judgments, that is to say,ai j a jk = aik for all i, j, k = 1, . . . , n, the measure proposed by Saaty (Saaty 1980) toevaluate the inconsistency of a pairwise comparison matrix, A(n × n) = (ai j ), called asSaaty’s Consistency Ratio, given by:

CR = CI

RI(n)

where CI = (λmax(A) − n)/(n − 1) = 1/(n(n − 1))∑

i �= j (ei j − 1); λmax is the principaleigenvalue of A = (ai j ), RI (n) is the random consistency index for matrices of order n,and ei j = ai jw j/wi is the error obtained when using ai j to estimate wi/w j .

Although Saaty and his school (Saaty 1990; Vargas 1994, 1997) repeatedly suggest theuse of the eigenvector method as the priorization procedure, during recent years, otherpriorization methods have been proposed in the literature. In what follows, and due totheir mathematical and psychological properties (Van den Honert 1998; Brugha 2000;Aguaron and Moreno-Jimenez 2000; Escobar and Moreno-Jimenez 2000), we will usethe raw geometric mean method (RGMM) as priorization procedure and the GeometricConsistency Index (GCI) as the inconsistency measure (Aguaron and Moreno-Jimenez2003).


According to this method, if A(n × n) = (ai j ) is the pairwise comparison matrix, thelocal priorities and the GCI are given, respectively, by:

wi =(

n∏

j=1

ai j

)1/n

GCI = 2

(n − 1)(n − 2)

∑

i< j

log2 ei j

AHP has been widely used in group decision making (Saaty 1989; Dyer and Forman1992; Iz and Gardiner 1993; Davies 1994; Ramanathan and Ganesh 1994; Bryson 1996;Bryson and Joseph 1999; Moreno-Jimenez et al. 2000). Two of the most extended methodsin this sense (Ramanathan and Ganesh 1994; Forman and Peniwati 1998) are: (i) the aggre-gation of individual judgments (AIJ) and (ii) the aggregation of individual priorities (AIP).

Let A[k] = (a[k]i j ) be the individual pairwise comparison matrix provided by the k-th

decision maker (k = 1, . . . , r ), and w[k] the respective individual priorities. Under the fisrtmethod (AIJ), a new pairwise comparison matrix for the group is constructed, AG = (aG

i j ),where each entry is the weighted geometric mean of the individual judgments for thesame entry. From this matrix, the priority vector is derived following any of the existingpriorization procedures.

When using the RGMM priorization procedure, the local priorities are given by:

ωGi =

(n∏

j=1

aGi j

)1/n

i = 1, . . . , n with aGi j =

(r∏

k=1

a[k]i j

)1/r

Under second method (AIP), the group priorities are given by:

wGi =

(r∏

k=1

w[k]i

)1/r

i = 1, . . . , n with w[k]i =

(n∏

j=1

a[k]i j

)1/n

It can be shown that the RGMM provides the same value for the AIJ and AIP approaches.However, this is not true for the eigenvector method (Escobar et al. 2004). In both situations(AIJ and AIP), individual positions are automatically aggregated even when they are insignificant conflict, a fact that usually results in an uncomfortable reaction on the part of theactors involved in the resolution process. Thus, some other ideas, such as the minimizationof the regret of the group (Escobar and Moreno-Jimenez 2000, 2002), and the search forthe core of consistency (Moreno-Jimenez et al. 2003) have been explored in AHP–groupdecision making.

Following this last approach, in the next section of the work we present a spreadsheetmodule developed with Visual Basic and Microsoft Excel, that allows us: (i) to calculate,for a fixed value of the inconsistency level, the preference structures associated with theinterval judgment matrix corresponding with the core of consistency of the group (CCM),and (ii) to analyse the evolution of the preference structures when changing the thresholdof inconsistency. This information will thereafter be used in the search for agreement in the


final ranking of alternatives between the actors in the negotiation process followed in theresolution of the problem.

In this sense, it is necessary to recall the following concepts and results, that can be seenmore in detail in Moreno-Jimenez et al. (2003).

Definition 1.

(1a) We define the Preference Structures Distribution (PS) associated to an interval judg-ment matrix, as the probability distribution of the n! possible ranking (preferencestructures) of the n alternatives.

(1b) For a fixed inconsistency threshold, we define the Consistency Stability Interval (CSI)of a judgment ars as the interval in which it can oscillate without the inconsistencymeasure of the matrix exceeding the threshold.

(1c) We define the Consistent Consensus Matrix (CCM) as the interval judgment matrixassociated with the core of consistency of the group. Each of the entries of this matrix,usually incomplete, represent the maximum range of values for this entry in which allthe actors are simultaneously consistent in their respective initial matrices.

Theorem 1. Let A = (ai j ) be an incomplete pairwise comparison matrix. Furthermore,let Pi be the set of vertices adjacent to a node i and Ni its degree (Ni ≥ 1). The prioritiesof an incomplete pairwise comparison matrix A = (ai j ) calculated by means of the rowgeometric mean priorization procedure, are obtained by solving

Ni log ωi −∑

j∈Pi

log ω j =∑

j∈Pi

log ai j (i = 1, 2, . . . , n) (1)

Theorem 2. Given a pairwise comparison matrix A = (ai j ), let GC I (A) be its incon-sistency evaluated by the GCI measure. If � is the oscillation allowed for this value, sothat the total inconsistency does not exceed a threshold previously fixed for the GCI, theconsistency stability interval for the judgment ars is given by [a

¯ rs, ars] = [arsδ¯ rs, ars δrs]

with

δ¯ rs(�) = en log ρmin

δrs(�) = en log ρmax

where [log ρmin, log ρmax] is the interval for log ρrs , with ρrs = (a′rs/ars)1/n, which satisfies

that:

2n

(n − 1)(n − 2)[(n − 2) log2 ρrs + 2 log ers log ρrs] ≤ � (2)

Theorem 3. Given a pairwise comparison matrix A = (ai j ) i, j = 1, . . . , n, the Recipro-cal Consistency Stability Interval for the judgment ars , when some judgments of the initial


matrix oscillate in interval judgments (a′i j ∈ Ii j , (i, j) ∈ C) is given by [ars/ed∗

, arsed∗]

where d∗ is the optimal value of the following optimization problem:

Min d = |xrs |s.t.

1

(n − 1)(n − 2)

[∑

(i, j)∈C

(log ei j + xi j )2 − 2

n

n∑

i=1

(n∑

j=1

xi j

)2]

≥ GC I + �

(3)a′

i j = ai j exi j ∈ Ii j for all (i, j) ∈ C

xi j + x ji = 0 (i, j) ∈ C

xi j = 0 (i, j) /∈ C

where C = C ∪ (r, s), and C = {(i, j), i < j, j ∈ Pi }.

Taking into account the previous results, the procedure which, based on the consistencyconcept, allows us to obtain the preference structures for a fixed GCI threshold, consistsof five steps: (i) initialisation; (ii) selection of the next entry to be included in the CCM;(iii) obtaining the CCM entries; (iv) simulation values for the CCM entries and calculationof the priorities using Tone’s procedure, and (v) deriving the preference structure of thealternatives. This procedure is supported by the following spreadsheet module.

3. The Spreadsheet Module

The module is written in Microsoft Visual Basic and is linked to a Microsoft Excel file. Themotivation for this choice is to make the module as accessible as possible. We have adoptedthis approach because the construction of the Consistent Consensus Matrix (CCM) requiresnon linear optimization, and it is not easy to distribute our module within commercialsoftware. Thus, we have used Microsoft Excel, which is available on most computers andcontains Solver, which is a good enough tool to perform optimization. To favour easyinteraction with the user, we work with Microsoft Visual Basic, given that it is a goodcoding environment to work with Microsoft Excel.

The structure of the module is shown in Figure 1. From the main window (Figure 2), weaccess to all the features of the module.

The central area of the Main Window is devoted to the visualization of the pairwisecomparison matrix corresponding to the different decision makers. The selection of a specificdecision maker can be made with the horizontal scroll bar. At the bottom of this windowthere are two command buttons. The left one leads to the Configuration Window (Figure 3)and the right to the Data Input Windows (Figure 4). The Configuration Window allows theuser to introduce the number of actors and the criteria. These parameters set up the dimensionof the problem and are required before introducing the judgments. A brief description ofthe criteria can also be entered in the Configuration Window. Any change made is saved


Figure 1. Module scheme.

Figure 2. Main window.

in a configuration file, so the last configuration will be available the next time the moduleruns.

In the Data Input Window the user, following the fundamental scale proposed by Saaty(Saaty 1980), can include the judgments of the pairwise comparison matrices correspondingto the evaluation of the alternatives for the different actors. First, the user must select thedecision maker whose matrix he wants to introduce. This selection edits the matrix and,picking on cell, we can now incorporate the judgment. Any change made is saved in a datafile, so, once again, the last data entered will be available the next time the module runs.

Returning to the Main Window, the right hand side includes the Start and End commandbuttons. The module offers two options to start: the first provides for a specific value of theinconsistency the preference structures (PS) derived from the CCM; the second providesthe evolution of these PS when changing the inconsistency thresholds.


Figure 3. Configuration window.

Figure 4. Data input window.

When the user clicks on any of these two command buttons, this begins the resolutionprocess that follows all the steps proposed in the previous section. For a fixed value of the CR,the module determines the order of the entries to be considered in the CCM construction and,for each entry of the CCM, the maximum interval in which all the actors are simultaneouslyconsistent in their respective initial judgment matrices (CSI). This calculation requires theuse of an optimisation algorithm, and this is precisely the moment when Microsoft Excel’sSolver starts working.

First, the data needed in the optimisation procedure, including the optimisation functionand the constraints, are introduced in a Microsoft Excel sheet. Thus, Solver is invoked andthe module automatically locates the cells which contain the function and the constraintsand gives them to Solver. The location of these cells can be different from one executionto another, because the dimensions of the problem (number of decision makers, number


Figure 5. Results windows.

of criteria) can be changed by the user. The whole process (filling in, setting the cells andcalling Solver) is required for each optimization. Although Microsoft Excel and Solver arecalled a number of times, these are hidden operations to the final user, who doesn’t see theMicrosoft Excel sheet at any moment in time.

After using Solver, the result of the optimisation is located in a Microsoft Excel cell.The value of this cell is taken back to the module, which uses it to calculate the consistencystability intervals. The algorithm continues until no more intervals can be added to theCCM without exceeding the inconsistency limit fixed in advance. From the resulting matrix(CCM), usually an incomplete one, the associated preference structures are obtained bymeans of simulation techniques and Tone’s algorithm (Theorem 1).

A summary of the results obtained in the module appears in the Results Window(Figure 5). In this window, the user can appreciate three different blocks: one for the InputData at the top of the window, and two other blocks at the bottom. The left hand sideshows the Consistent Consensus Matrix, whilst the right hand side presents the preferencestructures derived from the consistent consensus matrix.

In addition to this summary of results included in the Results Window of the program,more detailed numeric and graphic output are generated in a Microsoft Excel sheet. Usingthe previously created link between the module and Microsoft Excel, the numeric valuescorresponding to the CCM obtained for each CR, as well as their associated preferencestructures, are included in the Excel Results sheet (Figure 6).

The other sheet of the Microsoft Excel Results includes some graphic representationsof the evolution of the preference structures derived from the CCM when changing theinconsistency level (CR). Thus, Figures 7 and 8 correspond, respectively, to the valuepaths for preference structures and alternatives, whilst Figures 9 and 10 present their radialrepresentations.


Figure 6. Excel results sheet.

Figure 7. Value paths for PS.

Figure 8. Value paths for alternatives.


Figure 9. Radial graphic for preference structures.

Figure 10. Radial graphic for alternatives.

The Value Paths for Preference Structures (PS) shows the evolution, for different CRs,of the probability of each preference structure generated with the method, whilst the valuepaths for alternatives show the evolution for different CRs of the probability that eachalternative was selected as the best (greater priority).

Figures 7 and 8 represent, respectively, the evolution of all preference structure andalternatives, grouped all together in the same graphic. An individual visualization ofthis evolution can be seen in Figures 9 and 10. These Radial Graphic representationsmaps, respectively, every preference structure and alternative to a planar polygon whosevertices are placed at the end of rays cast from a central point. These rays themselvescorrespond to the different consistency ratios, and their length is proportional to theprobability of the associated preference structure or alternative. All the rays are equallyspaced, and the resulting polygon is obtained by joining every vertex to its neighbours.In both cases, that is to say, for preference structures and alternatives, the polygonsobtained are shown in a table which allows us to appreciate the evolution of all them at aglance.

4. Case Study

For a better understanding of the proposed spreadsheet module, in this section we apply it,first, to the same example considered in Moreno-Jimenez (2003), and, secondly, to a moregeneral case with two criteria.


Let us begin with the first of these two situations, where the following are the pairwisecomparison matrices for the two decision makers:

AI =

1 3 2 5

1/3 1 1/2 1

1/2 2 1 3

1/5 1 1/3 1

AI I =

1 1/3 1/2 2

3 1 1 4

2 1 1 3

1/2 1/4 1/3 1

In this case, the pairwise comparison matrix for the group, obtained by using the AIJmethod, is:

AG =

1 1 1√

10

1 1√

1/2 2

1√

2 1 3√1/10 1/2 1/3 1

and the priorities, the GCI’s and the rankings for the three matrices take the form as shownin Tables 1 and 2.

Following the five steps of the procedure, we have that:

Table 1. Individual and group priorities and GCI.

Alternative [I] [II] [G]

A 0.487217 0.163831 0.307379

B 0.133016 0.401302 0.251364

C 0.273982 0.337453 0.330814

D 0.105784 0.097414 0.110443

GCI 0.045112 0.041194 0.028021

Table 2. Rankings of alternatives.

Rank [I] [II] G

1 A B C

2 C C A

3 B A B

4 D D D


Step 1: The log A[k], k = I, I I are given by:

log AI =

0 1.099 0.693 1.609

0 −0.693 0

0 1.099

0

log AI I =

0 −1.099 −0.693 0.693

0 0 1.386

0 1.099

0

Step 2. The dispersion matrix for log A[k], k = I, I I is given by:

D =

− 1.207 0.480 0.210

− 0.120 0.480

− 0.000−

From this matrix, the order of the six entries to be included in the CCM is: (3,4), (2,3),(1,4), (1,3) or (2,4), and (1,2).

As the first two steps are not dependant on the CR, the order of the six entries will notdepend on the CR.

Step 3. This step concludes with the construction of the CCM. Depending on the selectedCR, we obtain a different number of non–null entries. When this number is less than 3(n −1 = 3), we artificially incorporate to the CCM the judgement of the AG correspondingto the selected cell. For 5% ≤ CR < 7%, the are only two non–null entries (aC

34, aC23). In

this case, we add to the CCM the value aG14 = 1/

√10. For 7% ≤ CR < 14%, the algorithm

provides three non–null entries (aC34, aC

23, aC14). For a CR greater than 14%, a fourth entry is

incorporated to the CCR, and so on.Steps 4 and 5 provide the evolution of the preference structures for different CR.For a better understanding of the procedure, let us take the case of CR = 7% (see Table 3).The first judgment to be included in the CCM, AC = (aC

i j ), is aC34 = 3, because its value

is the same for both decision makers. The next judgment to be included corresponds to the

Table 3. Values used in the construction of the CCM for CR = 7%.

Judgment’s incorporat. Initial Judgments CSI for k = I CSI for k = I I CSI for the group

Order Judgment Min Max Min Max Min Max Min Max

1 (3,4) 3.0000 3.0000 – – – – 3 32 (2,3) 0.5000 1.0000 0.1394 1.5940 0.3990 5.0121 0.5000 1.00003 (1,4) 2.0000 5.0000 2.5911 9.6483 1.5113 2.6468 2.5911 2.64684 (1,3) 0.5000 2.0000 1.2438 3.2160 0.5000 0.5000 – –5 (2,4) 1.0000 4.0000 – – – – – –6 (1,2) 0.3333 3.0000 – – – – – –


position (2, 3). The initial values for the two decision makers are aI23 = 0.5 and aI I

23 = 1.In this case, the consistency stability intervals (CSI) are, respectively:

[a¯

I23, a I

23

] = [0.1394, 1.5940] and[a¯

I I23, a I I

23

] = [0.3990, 5.0121]

Obviously, the intersection of these CSI and the interval for the initial values, coincideswith the last one, that is to say, aC

23 ∈ [0.5, 1].The next cell to be included in the CCM corresponds to the position (1, 4). Here, the

initial judgments are aI14 = 5 and aI I

14 = 2. For the CR = 7%, their respective CSI are:

[a¯

I14, a I

14

] = [2.5911, 9.6483] and[a¯

I I14, a I I

14

] = [1.5113, 2.6468]

and the actors’ core of consistency for this judgment (aC14), obtained by means of the

intersection with [2, 5], is the interval [2.5911, 2.6468].The fourth entry to be included in the CCM is the (1, 3) position. Now, the initial values

are aI13 = 2 and aI I

13 = 0.5, and the associated CSI are:

[a¯

I13, a I

13

] = [1.2438, 3.2160] and[a¯

I I13, a I I

13

] = [0.5000, 0.5000]

It can be seen that the CSI for the second decision maker is simply the initial judgmentthat does not belong to the CSI for the first decision maker. Thus, we cannot determine aninterval judgment where both actors are simultaneously consistent for a CR = 7%. So, theCCM has only three non–null entries. The CCM is given by:

CCM =

1 − − [2.59, 2.65]

1 [1/2, 1] −1 3

1

Using simulation techniques and Tone’s method, the preference structures probabilitiesfor CR = 7% are calculated (Table 4).

In closing this section, let us consider the construction of the CCM for a CR = 14% (seeTable 5).

The procedure is the same as for CR = 7%. The only difference, as can be seen inTable 5, is that the CSI for the CR = 14% are greater than for the CR = 7%. This results

Table 4. PS Probabilities for CR = 7%.

PS Probability

C–A–B–D 0.8585

C–B–A–D 0.1415


Table 5. Values used in the construction of the CCM for CR = 14%.

Judgment’s incorporat. Initial Judgments CSI for k = I CSI for k = I I CSI for the group

Order Judgment Min Max Min Max Min Max Min Max

1 (3,4) 3.0000 3.0000 – – – – 3.0000 3.00002 (2,3) 0.5000 1.0000 0.0843 2.6358 0.2446 8.1774 0.5000 1.00003 (1,4) 2.0000 5.0000 1.3583 18.4053 0.7202 5.5539 2.0000 5.00004 (1,3) 0.5000 2.0000 0.7655 5.2250 0.3107 0.8046 0.7655 0.80465 (2,4) 1.0000 4.0000 0.6219 1.6079 3.3108 4.8327 – –6 (1,2) 0.3333 3.0000 – – – – – –

in a new entry being incorporated into the CCM. In this case, the judgment aC13 belongs

to the interval [0.7655, 0.8046]. This fact provokes a drastic change in the ranks betweenalternatives A and C . Until CR = 13%, alternative A dominates alternative C but with theinclusion of the judgment aC

13, alternative C dominates alternative A (see Tables 6 and 7).This analysis can be extended by increasing the CR value. However, in practical situ-

ations, a CR less than 10% has been suggested (Saaty 1980). Having said that, when thecomplexity and the uncertainty of the problem increases, a CR of around 20% could beadmitted. Our work, following the cognitive constructivism approach proposed in Moreno–Jimenez et al. (1999, 2001), includes up to CR = 24%, where a new breakpoint is produced(see Tables 8 and 9).


PS Probability

C–A–B–D 0.2605

C–B–A–D 0.0752

A–C–B–D 0.6643


PS Probability

C–A–B–D 0.8137

C–B–A–D 0.1372

A–C–B–D 0.0491


PS Probability

C–A–B–D 0.3417

C–B–A–D 0.1365

A–C–B–D 0.5218



PS Probability

C–A–B–D 0.2800

C–B–A–D 0.2208

A–C–B–D 0.4304

A–B–C–D 0.0506

B–C–A–D 0.0148

B–A–C–D 0.0034

For CR = 14%, it can be seen that the only empty entries of the CCM are the cells (2, 4)and (1, 2). This means that the initial positions of the actors were in conflict for this entries.This fact can be taken into account in a second iteration of the negotiation process. Afterthe learning process derived from the resolution of the problem, the decision makers canreconsider their initial judgments for theses cells.

Turning now to the second of the two situations described at the beginning of this Section,we add a second criterion to the problem with the same decision makers being asked tobuild the individual judgment matrices for this second criterion:

B I =

1 5 1 5

1/5 1 1/3 2

1 3 1 4

1/5 1/2 1/4 1

B I I =

1 1/2 1/3 2

2 1 1 3

3 1 1 4

1/2 1/3 1/4 1

The preference structures obtained by repeating the previously explained method withthe judgement matrices of the new criterion are shown in Tables 10 and 11.

Assuming equal importance for each criterion (p = 0.5), the consensus multicreteriapreference structures are given in Tables 12 and 13. The graphic evolution of these preferencestructures can be seen in Figures 11 and 12.

Table 10. Probability of preference structures for the second criterion.

7% 8% 9% 10% 11% 12% 13% 14% 15%

C–A–B–D 0.9678 0.7384 0.6073 0.5285 0.5057 0.4999 0.4997

C–B–A–D 0.0103 0.0541 0.0920 0.1251 0.1664 0.1683 0.1699

A–C–B–D 0.0219 0.2075 0.3007 0.3464 0.3279 0.3318 0.3304

16% 17% 18% 19% 20% 21% 22% 23% 24%

C–A–B–D 0.9111 0.8880 0.8560 0.8155 0.8045 0.7992 0.7972 0.7956 0.7777

C–B–A–D 0.0889 0.1120 0.1440 0.1845 0.1955 0.2008 0.2028 0.2035 0.2223

A–C–B–D 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000


Table 11. Probability of being ranked first for the second criterion.

7% 8% 9% 10% 11% 12% 13% 14% 15%

A 0.0219 0.2075 0.3007 0.3464 0.3279 0.3318 0.3304

B

C 0.9781 0.7925 0.6993 0.6536 0.6721 0.6682 0.6696

D

16% 17% 18% 19% 20% 21% 22% 23% 24%

A 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

B

C 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

D

Table 12. Probability of preference structures for both criteria.

7% 8% 9% 10% 11% 12% 13% 14% 15%

C–A–B–D 0.8521 0.7744 0.7469 0.5761 0.4749 0.4083 0.3861 0.6592 0.5957

C–B–A–D 0.1479 0.1294 0.0671 0.0825 0.0944 0.1032 0.1196 0.1521 0.1424

A–C–B–D 0.0962 0.1861 0.3415 0.4308 0.4885 0.4944 0.1888 0.2619

A–B–C–D

B–C–A–D

B–A–C–D

16% 17% 18% 19% 20% 21% 22% 23% 24%

C–A–B–D 0.7602 0.7094 0.6664 0.6281 0.6026 0.5891 0.5721 0.3362 0.3318

C–B–A–D 0.1046 0.1174 0.1393 0.1641 0.1759 0.1750 0.1711 0.1314 0.1282

A–C–B–D 0.1353 0.1733 0.1944 0.2079 0.2215 0.2360 0.2568 0.1329 0.1405

A–B–C–D 0.0859 0.0865

B–C–A–D 0.2497 0.2436

B–A–C–D 0.0638 0.0694

Table 13. Probability of being ranked first for both criteria.

7% 8% 9% 10% 11% 12% 13% 14% 15%

A 0.0962 0.1861 0.3415 0.4308 0.4885 0.4994 0.1888 0.2619BC 1.0000 0.9038 0.8139 0.6586 0.5693 0.5115 0.5056 0.8113 0.7381D

16% 17% 18% 19% 20% 21% 22% 23% 24%

A 0.1353 0.1733 0.1944 0.2079 0.2215 0.2360 0.2568 0.2385 0.2478B 0.0064 0.0070C 0.8648 0.8267 0.8056 0.7922 0.7785 0.7640 0.7432 0.7552 0.7453D


Figure 11. Value paths for PS.

Figure 12. Value paths for alternatives.

5. Conclusions

Following the cognitive constructivism paradigm proposed in Moreno-Jimenez et al. (1999,2001), this paper presents a spreadsheet module that implements the consensus buildingapproach presented in Moreno-Jimenez et al. (2003) for group decision making in a localcontext with AHP.

This software allows us to obtain the preference structures derived from the ConsistentConsensus Matrix corresponding to the core of consistency of the group for a fixed CR. Thebehaviour of these preference structures when changing the inconsistency thresholds canbe used as the compromise solution for the group, when this is the way previously agreedto. Otherwise, the PS and, more particularly, the evolution of this PS when changing theCR, can be used as the starting point in any negotiated decision making process.

To that end, the actors involved in the resolution process could, in a second step, relaxtheir inconsistency levels to favour the search for agreement in negotiated decision making.


Furthermore, it is possible to limit the inconsistency consumed by some specific judgments,thereby allowing the incorporation of more entries into the CCM. Thus, as is the case withthe redundant judgments in AHP (Saaty 1980), we can obtain more accurate priorities.

From a practical point of view, and to the best of our knowledge, this is the first time thata software based on consistency has been developed to search for consensus in AHP groupdecision making. From a theoretical point of view, the underlying ideas upon which thesoftware is supported can be extended to a global context (hierarchy) and to any other mul-ticriteria decision technique, with the only peculiarity being the need to adapt the definitionof consistency. This is precisely what occurs with the software program presented here.

The proposed module has been developed with Visual Basic and Microsoft Excel in orderto make it as public and accessible as possible, in line with that claimed for the decisionaltools employed under the new scientific paradigms (Roy 1993).

The graphical output, combined with the interactivity of the Microsoft Excel graphics,results in a powerful tool for extracting information from the results and, therefore, forunderstanding the behaviour of the probability structures and their dependence on theconsistency ratios, as well as for exploring the interrelations between structures.

Acknowledgement

We wish to thank Stephen Wilkins for helping with the English translation of the text.

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A Spreadsheet Module for Consistent Consensus Building in AHP-Group Decision Making

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