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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 349729, 12 pages

http://dx.doi.org/10.1155/2014/349729

## Characterization of Consistent Completion of Reciprocal Comparison Matrices

^{1}Instituto de Matemática Multidisciplinar (IMM), Universitat Politècnica de València, Camino de Vera S/N, 46022 Valencia, Spain^{2}Universitat Politècnica de València, Camino de Vera S/N, 46022 Valencia, Spain^{3}Fluing IMM, Universitat Politècnica de València, Camino de Vera S/N, Edificio 5C bajo, 46022 València, Spain

Received 22 October 2013; Revised 6 December 2013; Accepted 6 December 2013; Published 25 February 2014

Academic Editor: L. Jódar

Copyright © 2014 Julio Benítez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Analytic hierarchy process (AHP) is a leading multi-attribute decision-aiding model that is designed to help make better choices when faced with complex decisions involving several dimensions. AHP, which enables qualitative analysis using a combination of subjective and objective information, is a multiple criteria decision analysis approach that uses hierarchical structured pairwise comparisons. One of the drawbacks of AHP is that a pairwise comparison cannot be completed by an actor or stakeholder not fully familiar with all the aspects of the problem. The authors have developed a completion based on a process of linearization that minimizes the matrix distance defined in terms of the Frobenius norm (a strictly convex minimization problem). In this paper, we characterize when an incomplete, positive, and reciprocal matrix can be completed to become a consistent matrix. We show that this characterization reduces the problem to the solution of a linear system of equations—a straightforward procedure. Various properties of such a completion are also developed using graph theory, including explicit calculation formulas. In real decision-making processes, facilitators conducting the study could use these characterizations to accept an incomplete comparison body given by an actor or to encourage the actor to further develop the comparison for the sake of consistency.

#### 1. Introduction

The so-called analytic hierarchy process (AHP) [1, 2] has been accepted as a leading multiattribute decision-aiding model both by practitioners and academicians, since it is designed to make better choices when faced with complex decisions involving several dimensions. As a multiple criteria decision analysis (MCDA) technique [3], AHP solves optimization discrete decision problems that involve choosing one of several alternatives. In many fields, decision making (DM) has become very complex due to a large number of alternatives and multiple goals that sometimes conflict with each other. The AHP approach, which enables qualitative analysis using a combination of subjective and objective information/data, is an MCDA approach that uses hierarchically structured pairwise comparisons.

One of the weaknesses of AHP, which AHP shares with other decision models, comes from the fact that typically the input is static. In other words, users must provide all the preference data at the same time, and the criteria must be completely defined from the start. Nevertheless, changing scenarios are currently more than frequent, due to various sources of uncertainty, and pairwise comparison cannot be successfully completed when there are many alternatives to be considered and/or the comparison is required from an actor or stakeholder not fully familiar with all the aspects of the problem. The current trend toward greater interactive involvement of citizens in policy making is unavoidable and highly desirable. It is generally agreed that better decisions are implemented with less conflict and more success when they are driven by stakeholders [4]. Public participation is also likely to improve the quality of decisions; using a wider pool of knowledge and understanding can prevent obstacles that would obstruct effective implementation of a particular decision [5, 6].

For example, environmental (field of expertise of the authors) projects and programs are likely to be more relevant, successful, and sustainable if their actors are involved in planning, implementation, and evaluation [7, 8]. Moreover, integrating water resource management is a reference framework for water management in many countries [9]. For example, the Water Framework Directive (WFD) [10] was enforced in Europe in 2000. Taking the example down a level in concretion, in [11] the authors describe a management model to address the main needs of an aquifer, with this being a problem of environmental, social and public policy; the relevant decisions are enriched, when incorporating the interests of the parties concerned, including academics, users, and administrators of the aquifer.

However, some actors may not be completely familiar with one or more of the elements about which they have to issue their judgement or opinion. As a result, it is difficult to gather complete information about the preferences of such a stakeholder at a given moment. It seems reasonable to enable such actors to express their preferences several times at their own convenience. Meanwhile, partial results based on partial preference data may be generated from data collected at various times—and this data may eventually be consolidated when the information is complete. Based on a process of linearization [12] that minimizes a matrix distance defined in terms of the Frobenius norm (a strictly convex minimization problem based on the best one rank approximation) in [13] the authors have initiated a line towards a dynamic model of AHP by addressing the problem of adding new criteria or deleting obsolete criteria. In [14] this research line was extended to propose a framework that enables users to provide data on their preferences in a partial and/or incomplete way and at different times. The consistent completion of a reciprocal matrix as a mechanism to obtain a consistent body of opinion issued in an incomplete manner by a specific actor was addressed. This feature can help stakeholders not fully problem acquainted participate in processes.

Public participation is, however, not a panacea. Collaboration and participation cannot solve every problem and should not be used as a surrogate for other systematic attempts to plan and manage decision issues [15, 16]. Public participation efforts must be responsive to the needs of the stakeholders. It is also critical to recognize that participation processes require a flexible approach that is appropriate to user conditions. As a result, the need to integrate multiple criteria and uncertainty demands a systematic framework to suitably represent and handle the information [17].

In particular, uncertainty produced by a lack of comprehensive knowledge on the part of the stakeholders must be handled suitably. In a real participatory DM process, the facilitator in charge of conducting the study may have to face an incomplete body of opinion given by an actor. In this event, he or she needs robust criteria to either accept the opinion or encourage the actor to further develop the comparison so that the judgment is eventually completed consistently, thus helping ensure an optimal decision.

In this paper, we provide a solution to this issue by solving the following problem: to characterize when an incomplete, positive, reciprocal matrix can be completed to become a consistent matrix. If the incomplete judgment given by an actor passes this test, then the facilitator will feel confident about accepting it. Otherwise, the facilitator should keep on working with the actor until the characterization criterion is met. We show that this characterization reduces the consistent completion of an incomplete, positive, reciprocal matrix to the solution of a linear system of equations—a straightforward procedure. In addition, the uniqueness of the completion is studied using graph theory and we offer several ways to find such completion when it exists.

This paper is organized as follows. In Section 2 we give a formal statement of the problem and provide some prerequisites. Sections 3 and 4, respectively, address the main objectives of the paper, namely, the characterization and the uniqueness, together with the computation of the consistent completion (when it exists). In Section 5, these results are applied to two comparison matrices for deciding suitable leakage policies within the framework of water supply management. Section 6 contains the conclusions and closes the paper.

#### 2. Prerequisites and Formal Statement of the Problem

##### 2.1. A Brief Review of AHP

AHP [1, 2, 18] is a multiple attribute decision method that uses structured pairwise comparisons with numerical judgements from an absolute scale of numbers. The fundamentals of AHP, including its hierarchical, multilevel structure with goals, criteria, and alternatives, the way in which judgement is compiled into positive reciprocal matrices, the estimation of the relative weights of the decision elements, the use of prioritization techniques, and the way in which aggregation is performed to obtain a final composite vector of priorities, can be found in any handbook and many papers about the subject (see, e.g., [1, 17, 18]).

In this paper, we consider a nine-point scale developed by Saaty [1, 2], with the possibility of including intermediate numerical values on this scale. Thus, as a result of the comparison performed, an matrix , is formed, with being the number of the decision elements and measuring the relative importance of element over element . To extract priority vectors from the comparison matrices, the eigenvector method, which was first proposed by Saaty in his seminal paper [1], is used in this paper.

A comparison matrix, , exhibits a basic property, namely *reciprocity*:
This property implies the *homogeneity*: if elements and are considered equally important, then . In particular, one has .

*Definition 1. *An matrix is *reciprocal* when for any and the condition (1) is satisfied.

In addition to the reciprocity property, another property, consistency, should theoretically be desirable for a comparison matrix.

*Definition 2. *An matrix is consistent when for any and
is satisfied.

Consistency expresses the coherence that may exist between judgements about the elements of a set. Since preferences are expressed in a subjective manner it is reasonable for some kind of incoherence to exist. When dealing with intangibles, judgements are rarely consistent unless they are forced in some artificial manner.

In addition, a comparison matrix is not generally consistent because it contains comparison values obtained through numerical judgement using a fixed scale. For most problems, estimates of these values by an expert are assumed to be small perturbations of the “right” values. This implies small perturbations for the eigenvalues and eigenvectors (see, e.g., [19]). For a consistent matrix , the leading eigenvalue and the principal (Perron) eigenvector of provide information to deal with complex decisions, with the normalized Perron eigenvector giving the sought priority vector [18]. In the general case, however, as said, is not consistent. For nonconsistent matrices, the problem to solve is the eigenvalue problem , where is the unique largest eigenvalue of that gives the Perron eigenvector as an estimate of the priority vector.

##### 2.2. Notations and Basic Facts

The set of real matrices is denoted by . We write . If is a matrix, then and will denote the trace and the transpose of , respectively. The standard basis of is denoted by . Any vector of will be considered a column. The vector will be denoted by .

As can be seen from (1) and (2), any consistent matrix is reciprocal. Also, it can be easily proven that the rank of any consistent matrix is 1.

We will use the mappings and given by and , respectively, where . Evidently, for , The image by of the set of consistent matrices will play an important role in the sequel. Precisely, we define A basic property of is established in the next result.

Theorem 3 (Theorem 2.2 of [12] ). *If we define
**
then is linear, , , and .*

##### 2.3. Problem Definition

The purpose of this paper is to characterize when a reciprocal and incomplete matrix can be completed to become a consistent matrix. We state the precise formulation of the completion problem.

*Problem 4. *Let and be such that for . Let be given positive numbers. Can we find a consistent matrix such that and for any ? In case that the answer to this question is affirmative, find all matrices verifying the condition of the aforementioned question.

Although the following result can be dealt by means of the general characterization given in Theorem 7, we will prove it using Theorem 3.

Theorem 5. *Let . The following statements are equivalent: * (i)*There exist , , and such that is consistent.* (ii)* is consistent.*

*Proof. *(i) (ii): since is consistent, the relations given in (2) hold for all indices and in particular for a subset of them.

(ii) (i): if is consistent, then, by Theorem 3, there exists such that . Let be an arbitrary vector of , and define in such a way that . Let us define
We get
Hence , showing that is consistent.

Observe that the proof of the former theorem enables us to give the general solution of the following problem: find all consistent completions of the matrix where is consistent. Since is consistent, there exists such that . It is now enough to pick any and define as in (6).

To motivate the notation and the precise definition of the problem considered in this paper, let us consider the following example. Let By taking logarithms of the entries of the matrix, the aforementioned completion problem can be managed. Since the image by of any reciprocal matrix is skew symmetric, in order to find a consistent completion of an incomplete reciprocal matrix, it is enough to restrict ourselves to the subset of reciprocal matrices of order . From (9) we obtain then any skew symmetric completion of is of the form where .

From now on, we define for the following skew symmetric matrices Thus, with this notation, the skew symmetric completion considered in equalities (9), (10), and (11) takes the simpler form where Furthermore, observe that matrix appearing in (13) can be written as where and are real numbers that can be easily determined from the incomplete matrix given in (9). In an informal way, we can think of as the incomplete skew symmetric matrix to be completed, and , and —and their symmetric positions with respect to the principal diagonal—as the void positions that must be filled.

We need the following lemma [14].

Lemma 6. *Let the matrices be defined as in (12) and let the mapping be defined as in (5). Then
**for any .*

#### 3. Characterization of the Completion of a Reciprocal Matrix

Now we are ready to establish the first main result of this paper. In the statement of the next theorem we use two sets of indices of matrix , , and . The unspecified entries of above the main diagonal will be located at the indices belonging to . For example, for the matrix given in (9) we have and .

Theorem 7. *Let be indices such that for . Denote and . Let . The following statements are equivalent.* (i)*There exist such that .* (ii)*There exists such that and for any .* (iii)*There exists such that for any .** Furthermore, in the case that the statements hold, then
*

*Proof. *(i) (ii): by Theorem 3, there exists such that
Note that any vector and in particular can be splitted as , where and is orthogonal to . Here we assume that the inner product in is the standard inner product; that is, . Hence, if , then . By Theorem 3 we obtain . Let us recall that the trace is a linear operator. If , then (18) and Lemma 6 yield
Let . We obtain from (18) and Lemma 6

(ii) (iii) is trivial.

(iii) (i): let us define for any and
Observe that from Lemma 6 one has
Let us now recall the following.(a) is skew symmetric. This fact can be easily deduced from (5) by proving . Hence (21) and the fact that any is also skew symmetric prove that is skew symmetric.(b)The set can be endowed with the inner product .(c)The matrices for form a basis for the subset of composed of skew symmetric matrices.

Previous item (c) and implications (22) lead to the fact that is orthogonal to any skew symmetric matrix. By previous item (a) we obtain . Thus, . Hence, by Theorem 3, we obtain .

*Note*. Observe that we have reduced the completion problem to study the linear system occurring in item (ii)—or (iii)—of the previous theorem. We shall see how this theorem works in two specific situations.

*Example 8. *We will apply Theorem 7 in order to show that matrix in (9) cannot be completed to be consistent. If this completion was feasible, then by Theorem 7 and (14), there would exist such that
It can be quickly shown that this linear system has no solution.

*Example 9. *We will see if
has a consistent completion. By taking logarithms, we construct
If there is a consistent completion, by item (iii) of Theorem 7, then there will exist such that
This system, clearly, is solvable. Hence, the completion is possible. We will see how Theorem 7 enables us to find such completion(s). The general solution of (26) is
If is any consistent completion of , then item (i) of Theorem 7 guarantees that there exists such that , and such can be obtained from (17) obtaining . Thus, . We conclude that there is a unique consistent completion of and it is given by

#### 4. Completion of Reciprocal Matrices and Graph Theory: Uniqueness of the Completion

In this section we develop several useful results that enable us to study the uniqueness of the consistent completion and to compute in a straightforward manner all possible completions. Let us start by having a deeper look at the linear systems (23) and (26).

We have, respectively, The transposes of the matrices of the above systems are the incidence matrices of the directed graphs of Figure 1.

For an arbitrary incomplete reciprocal matrix , we use the following procedure to construct a directed graph, denoted by : If , then there is no arrow from to . If and we do not know the entry , then there is no arrow from to . If and we know the entry , then there is an arrow from to . Now, we easily construct the incidence matrix of , denoted in the sequel by . Observe that any incidence matrix has exactly two nonzero entries, namely, and , in each column.

To describe the linear system that appears in item (iii) of Theorem 7, we also define, for an incomplete reciprocal matrix , the vector by the following procedure.(a) is the number of columns of ; that is, is the number of arrows of the directed graph . Explicitly, , where the meaning of is given in Theorem 7: is the number of entries of to be filled.(b)For , let us pay attention to the th column of and let be the unique indices such that the entry of is 1 and the entry of is −1. We set .

Observe that and are unique up to a permutation.

Theorem 7 can be rephrased as follows.

Theorem 10. *If is an incomplete reciprocal matrix, then can be completed to be a consistent matrix if and only if the system is consistent.*

We present here an m-file that can be executed in Matlab or Octave. This file checks if an incomplete reciprocal matrix can be completed to be consistent. We use Theorem 10 and the following criterion: the linear system is solvable if and only if , where the superscript means the Moore-Penrose inverse (see Algorithm 1).

There is a trivial bound for (let us recall that is the number of entries of to be filled). Since the diagonal entries of must be equal to 1, then . The following result gives a sufficient condition for a reciprocal incomplete matrix to be completed to be consistent.

Theorem 11. *Let be a reciprocal incomplete matrix and let be the number of void entries (located up and down the main diagonal of ). If has connected components and , then can be completed to be consistent.*

*Proof. *We denote and we construct as the previous procedure states. Evidently, . Since has connected components, then the rank of is (see, e.g., [20, Th. 2.3]). Let be the range space of , which obviously satisfies . From we obtain ; hence , which shows that the linear system is consistent.

*Example 12. *In this example we will see that the graph can be disconnected. Let be positive, and let us consider
The graph and its incidence matrix are shown in Figure 2. Obviously, has two connected components.

To find all possible consistent completions of , we must consider the system :
Its general solution is , .

If is any consistent completion of , then Theorem 7 ensures that and are such that
By denoting we obtain
Observe that the consistent completion of is not unique since is arbitrary.

In the case that the completion is possible, we will show that it is unique in a particular situation: the corresponding graph is connected. We will denote by the null space of matrix , that is, all vectors such that . For future use, we will need the following lemma.

Lemma 13. *Let be a graph and its incidence matrix. Then
**
Hence, the dimension of is the number of connected components of graph . In particular, if the graph is connected, then .*

*Proof. *Let be such that . This implies that “if there is an edge connecting nodes and , then .” Hence if and only if and are in the same connected component of .

In the next result, the hypothesis serves to guarantee .

Theorem 14. *Let be a reciprocal incomplete matrix and be the number of void entries (located up and down the main diagonal of ). If , is connected and there exists a consistent completion of , then this completion is unique.*

*Proof. *Let us define , , and as in Theorem 11. Since there is a consistent completion of , then the linear system is consistent. Let be a solution. It is simple to prove that . Using this fact and Lemma 13 we have that if is any solution of , then there exists such that
Any possible completion of must be computed using the values from (17); then for . This shows that the completion is unique.

Observe that if there is a consistent completion of , then the general solution of is given by , where is a particular solution of . It is simple to prove that if is a matrix satisfying , then verifies the system . Hence the general solution of this latter system is where is an arbitrary but fixed matrix such that . We can choose , where the superindex means the Moore-Penrose inverse of a matrix. Now, observe that and in [20, Ch. 2] a method for finding the Moore-Penrose inverse of an incidence matrix is given.

Another result that will be useful is the following (see [21, Ch. 2]). “Let be an matrix and such that the system is consistent. If is any matrix satisfying , then the general solution of the is given by for arbitrary ”. This result enables us to find the general solution of without computing the null space of as in (36).

Finally, notice that, to find the consistent completion of (as distilled in the proof of Theorem 14) when the corresponding graph is connected, we can discard the vector in the null space of appearing in (36).

*Example 15 (this is the revisited previous Example 9). *Let be the incomplete matrix given in (24). Following the notation of Table 1 we have , , , and . By Theorem 11 we obtain that there is a consistent completion. A solution of is (by employing as the Moore-Penrose inverse of )
and . This example finishes as before.

*Example 16 (this is the revisited Example 12). *Let be the incomplete matrix given in (30). Following the notation of Table 1 we have , , , and . Any solution of is given by
with . But any vector of is of the form , where . Hence
Theorem 5 ensures that and satisfy that is a matrix such that is any consistent completion of . By denoting we obtain the same solution of the Example 12 obtained previously.

Another way of obtaining the same solution is by means of
Obviously, one obtains the same solution as in (39) by assigning and .

Let us observe that the linear system is consistent if and only if . But standard linear algebra produces . Hence we have that the linear system is consistent if and only for any .

In next result we find the null space of for some types of graphs. To this end, we recall the concept of *cycle* in a graph. A cycle is a chain starting at a point and finishing at the same point. For instance, the left graph of Figure 1 has a cycle (observe that the edge is oriented in the opposite direction).

Let be an oriented graph (with the notation established in Table 1) whose edges are denoted by , with the enumeration of the edges being dictated by the incidence matrix. For example, for the left graph of Figure 1, the edge is the third one. For a of set of edges (maybe the orientation of these edges can be distinct to the one given by the graph) we define the following vector : set if and the orientation of agrees with the corresponding one of . Set if and the orientation of is the opposite of the corresponding one of . Set if . For example, for the left graph of Figure 1 we associate the cycle to because the first and third edges ( and ) are in the loop; the second edge () appears in the cycle but in reverse and the fourth edge () does not appear in the cycle.

Some further properties of the consistent completion of can be deduced if the associated graph is planar (see, e.g., [22] or any textbook on graph theory for a deeper insight of planar graphs). Let us remark that is not necessarily planar as the following example shows. Let
with being positive numbers. The matrix leads to a nonplanar graph, namely, the complete bipartite graph. Also, observe that a planar graph divides the plane into several *faces*; there is only one unbounded face; and any bounded face corresponds to a cycle. If denotes the number of bounded faces, the very well-known Euler’s formula holds:
It is a known fact (see, e.g., [22, Ch. 2]) that any vector corresponding to a cycle belongs to the null space of the incidence matrix of the graph.

Theorem 17. *Let be a planar oriented graph and its incidence matrix. If correspond to the bounded faces of the graph, then is a basis of .*

*Proof. *By the previous paragraph we have . We shall prove the next affirmation by induction on . “If correspond to the bounded faces of a graph, then are linearly independent”. For the theorem obviously is true since . Assume that the claim is true for and let correspond to the bounded faces of a graph such that . There must be at least one edge from the unbounded face. This edge, let us say the th edge, belongs to only one face (any edge is a boundary of two faces; but if one face is unbounded, the other face must be bounded), let us say this face. Looking at the th coordinate of we conclude that . Now, the faces of the graph obtained by deleting the th edge in the original graph are . Since , by the induction hypothesis we have that .

To finish the proof of the theorem, let us recall that the rank of the incidence matrix is , where is the number of points and the number of connected components. It will be also useful to remark that the incidence matrix has rows and columns, with being the number of edges. By Euler’s formula (42) we have
Since are independent vectors in a -dimensional subspace, these vectors form a basis of this subspace.

Corollary 18. *Let be an incomplete reciprocal matrix. If is planar and has no bounded faces, then there exists a consistent completion of . Observe that by construction, is always oriented.*

*Example 19 (Example 8 revisited). *Looking at the left graph of Figure 1, we see that the only bounded face corresponds to the cycle . Hence a basis of is . Since the matrix of Example 8 has a consistent completion if and only if , , and , matrix cannot be completed to a consistent matrix.

Let us now consider an incomplete reciprocal matrix that cannot be completed to be consistent. How can the known entries be modified to complete the matrix to be consistent? The answer will be clear if we recall the following summary.

For an incomplete reciprocal matrix , the following affirmations are equivalent. (i)There exists a consistent completion of . (ii)The linear system is consistent. (iii) for any .

*Example 20. *If we want to modify some entries of matrix given in (9) in order to have a consistent completion, let us start by writing
Now we can choose the entries by using one of the above items. But as we know the null space of —in fact, from Example 19 we know that a basis of is spanned by —we choose item (iii).
It is noteworthy that the value of is arbitrary.

#### 5. Application to Leakage Control in a Water Supply System

By way of illustration, in this section we use some examples corresponding to a problem that challenges water supply managers, and in which the main objective is to achieve a suitable leakage policy to minimize water loss (see, e.g., [23]). The criteria considered are given in the examples. Finally, various management alternatives for leakage control are considered: ranging from active leakage control (ALC) to passive leakage control (PLC). ALC involves taking actions in supply systems or individual hydrometric district areas, to identify and repair detectable leaks that have not been reported. PLC refers to repairing only reported or evident leaks.

We use the characterization presented in this paper as a decision support tool to assess the consistency of incomplete judgments given by specific stakeholders.

*Example 21. *We consider here the following multiple criteria to decide on the alternatives: C_{1}: planning development cost and its implementation, C_{2}: budgets and payment appropriations, C_{3}: investment recovery, C_{4}: social cost, C_{5}: environmental cost.

In this example, we use the point of view of an employee who develops water supply projects in a company in Spain. The actor has given an incomplete body of opinion and is unable to provide a robust viewpoint regarding some criteria, particularly comparisons regarding social () and environmental () costs, as well as planning development cost ().

Let be the following incomplete reciprocal matrix that represents the incomplete judgment given by the stakeholder:

By running the file given in Section 4, we find that there is no consistent completion of . However, the associated directed graph of this incomplete matrix is relatively easy to obtain (see Figure 3). We can see that this graph is planar. Thus, Theorem 17 enables us to very quickly obtain a basis of : the basis corresponds to cycles , , and . Explicitly we have that the following vectors form a basis of .

In this case, a slight modification of some entries of makes a consistent completion feasible. In effect, by forcing , we find that the three following conditions must be simultaneously satisfied: Observe that the first and the third equalities from (48) are satisfied by the original matrix , whereas the second equality does not hold.

The stakeholder considers that some entries can be changed. Since and appear in the first and third equalities, the stakeholder is asked to modify only , and his answer is affirmative. By modifying entry to , we obtain a matrix that is consistently completed. This alteration does not change the opinion of the stakeholder, because the old value, 7, and the new value, 8, correspond to similar verbal judgements of preference in Saaty’s scale. Let us rename , the new reciprocal incomplete matrix,

We execute the m-file in order to find , , and . This m-file also ensures that there is a consistent completion of (this fact was previously known because (48) are satisfied). This completion is unique by Theorem 14. As we mentioned, this unique completion of can be obtained from (36) by discarding the vectors of . By executing

≫w = N*b

where and , we obtain

w = 0.0000 0.6931 0.0000 −1.3863 0.6931

Theorem 7 enables us to obtain the unique completion. It can be quickly found by using the calls in Algorithm 2.

Thus obtaining a consistent completion of the matrix which is satisfactory for the stakeholder ≫A A = 1.0000 0.5000 1.0000 4.0000 0.5000 2.0000 1.0000 2.0000 8.0000 1.0000 1.0000 0.5000 1.0000 4.0000 0.5000 0.2500 0.1250 0.2500 1.0000 0.1250 2.0000 1.0000 2.0000 8.0000 1.0000

*Example 22. *A final example shows an incomplete judgment that cannot be consistently completed. The criteria used to decide between the alternatives may be varied and take into account economic, environmental, and social impacts. In addition to the five criteria shown in Example 21, two more criteria are used: C_{6}: construction of tanks and reservoirs, C_{7}: emissions.

Let be the matrix that corresponds to an incomplete judgment. By running the previous file, we find that there is no consistent completion of . We can obtain the directed graph corresponding to this example. This graph is more complex than the other example. However, since we know the entries for and , then the graph associated with the matrix of this example contains the complete bipartite graph. Hence, the graph associated with is not planar. In this case, the facilitator should invite the stakeholder to work further in order to reach a point where a consistent completion is possible.

#### 6. Conclusions

MCDA, in particular AHP, has emerged as a decision support tool to integrate technical information and stakeholder values. Over the last decade, there has been a significant growth in MCDA applications in many fields, including decision support tools [24]. These techniques provide a systematic approach to combine information inputs with benefit/cost information and decision maker or stakeholder views to rank alternatives [25].

Since vast investments are frequently at stake, DM must be performed with extreme care. In this regard, facilitators conducting participatory processes need robust tools that enable them to make discerning decisions when collecting opinions from various stakeholders.

In this paper, by characterizing incomplete comparison matrices that can be consistently completed we have provided a robust tool to decide if an incomplete comparison body given by an actor is acceptable or, on the contrary, the actor must further develop the comparison for the sake of consistency. This tool is very simple since it reduces to the solution of a linear set of equations. Using graph theory we have developed several useful results that enable us to study the uniqueness of the consistent completion and to compute all the possible completions in a straightforward manner. Finally, in Section 5 we have provided an illustrative example corresponding to the (incomplete) opinion given by a stakeholder in a DM process regarding the most suitable water leak management policy in a water distribution system. As can be observed, the calculations are simple and clear and can be easily implemented in any decision support tool based on AHP.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work has been supported by project IDAWAS, DPI2009-11591, of the Spanish Ministry of Science and Innovation, with supplementary support from ACOMP/2011/188 of the Education Department of the Generalitat Valenciana. The first two authors wish to thank the Universitat Politècnica de València for the support through the project PAID-06-12. The English language in this paper was revised by John Rawlins.

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