Derivation of Priorities and Weights for Set-Valued Matrices Using the Geometric Mean Approach

Abstract

Priorities are essential in the analytic hierarchy process (AHP). Several approaches have been proposed to derive priorities in the framework of the AHP. Priorities correspond to the weights in the weighted mean as well as in other aggregation operators as the ordered weighted averaging (OWA) operators, and the quasi-arithmetic means.

Derivation of priorities for the AHP typically starts by eliciting a preference matrix from an expert and then using this matrix to obtain the vector priorities. For consistent matrices, the vector of priorities is unique. Nevertheless, it is usual that the matrix is not consistent. In this case, different methods exist for extracting this vector from the matrix.

This article introduces a method for this purpose when the cells of the matrix are not a single value but a set of values. That is, we have a set-valued preference matrix. We discuss the relation of this type of matrices and hesitant fuzzy preference relations.

INTRODUCTION

Aggregation operators are used to fuse information proceeding from different information sources. There is a large number of these operators, the weighted mean being one of the most well known. For detail on aggregation operators see, for example, Torra and Narukawa (2007) and Beliakov, Pradera, and Calvo (2008).

Aggregation operators often rely on a parameter that permits us to supply some background knowledge on the data sources being aggregated. This is the case of the weighted mean, in which the weighting vector is used to represent the importance of the information sources. For example, the importance of the criteria in a multicriteria decision-making problem, or the importance of the experts in a group decision-making problem.

Quasi-arithmetic means, a family of functions that generalize the weighted mean, depend also on a weighting vector. This vector also represents the importance of the sources.

Other operators that also depend on a single weighting vector are the ordered weighted averaging (OWA) operator (Yager 1988) and its variations, for example, the family of quasi-OWA operators and the induced OWA. In this case, the weighting vector represents the relative importance of the values. For example, we can represent that we prefer low values or, to the contrary, that we prefer high values. Another operator that depends on two weighting vectors is the weighted ordered weighted averaging (WOWA) operator (Torra 1997). One of the vectors establishes the importance of the information sources and the other the importance of the values.

There exist several open problems in the field of aggregation operators. See, for example, Torra (2005) and Mesiar and Stupnanova (2015). One of them is the determination of the model. Once the aggregation operator is chosen, the model is completed by selecting the parameters of the operator. Nevertheless, this is not an easy task. There are several approaches in the literature for this problem. We distinguish two classes following Torra and Narukawa (2007).

• Methods based on an expert. An expert is interviewed and gives relevant information that is later used to extract the parameters. The information from the expert can take different forms.

  The approaches related to the analytical hierarchy process (AHP) (Saaty 1980; Saaty 1987) to extract weights for the weighted mean fall into this class. Some of the approaches for OWA weight determination also fill in this class.

  In the case of the AHP, it is usual to elicit from the expert a preference matrix and then use this matrix to find the weights of the weighted mean. In the case of the OWA, we ask for an orness level (Dujmovic 1973) and then we use this orness level to find the OWA weights.

• Methods based on data. Some examples or, in general, data are available and can be used to determine the parameters. Optimization and machine learning approaches are used in this context. For example, when examples are of the form (input,output), we can use them to fit a model based on a given aggregation operator. Unsupervised machine learning algorithms have also been used for this purpose when raw data is available (i.e., without an output variable). Alternatively, we may have a rank of data and determine the weights to approximate the rank.

See Torra and Narukawa (2007) for details on this classification and a description of some of the methods.

In this study, we consider the problem of determining the weights when we elicit a matrix from an expert (or a set of experts) and each cell in the matrix is not a single value but a set of values. This problem is inspired by hesitant fuzzy sets (Torra 2009; Torra and Narukawa 2009), in which, instead of a single membership value we have a set of them.

A preliminary version of this work was presented in Torra (2014), where we introduced this type of matrices and a solution for the determination of priorities. A matrix of this type is similar to the hesitant fuzzy preference relations (HFPR) and the hesitant multiplicative preference relations (HMPR; Xia and Xu 2013; Zhang and Wu 2014; Zhu and Xu 2014; Xu 2014). In these works, values in the matrix are typically limited to the interval because possible values are inspired in Saaty’s work (Saaty 1980, 1987). As we will discuss later, HMPR is defined so that if the matrix contains a for cell (i, j), then is contained in cell (j, i). In our definition, we have fewer constraints on the range of the values and on the values in each cell. It is important to note that in our matrices we do not have membership values because when , then does not belong to [0, 1].

The literature reviews that when matrices are elicited from an expert they are usually nonconsistent (Crawford and Williams 1985). To deal with this problem, methods are developed to determine the weights from a nonconsistent matrix. In this work we focus on the determination of the weights when the matrix is a set-valued matrix and it is not consistent.

The structure of the paper is as follows. In “Preliminaries,” we first review some results on aggregation operators and then the approach applied in the AHP to derive the weights. In “The Geometric Mean Approach for Set-Valued Matrices,” we present our approach and an example. We compare our solution with that in Zhang and Wu (2014). We discuss the consistency requirements of the set-valued matrices and propose an algorithm to make set-valued matrices consistent. The solution we propose is based on the properties that we require the function to satisfy. The article ends with some conclusions and lines for future research.

PRELIMINARIES

In this section we review a few results on aggregation operators and the AHP for the weighted mean.

Aggregation Operators

An aggregation operator is a function that satisfies unanimity and monotonicity. A detailed discussion of these functions can be found in Torra and Narukawa (2007) and Beliakov, Pradera, and Calvo (2008).

We consider aggregation operators that take n parameters in . We will use the notation to represent this function applied to values .

Let us start by considering separable functions .

Definition 2.1

is separable when there exist functions and ○ such that

with ○ a continuous, associative, and cancellative operator.

Let us now consider two basic properties for aggregation operators. First, unanimity or idempotency. An aggregation operator satisfies unanimity when the following equation holds:

(1)

Second, reciprocity. An aggregation operator satisfies reciprocity when the following equation holds:

(2)

The following result characterizes aggregation operators, which are separable and satisfy unanimity and reciprocity.

Proposition 2.2

(Aczel and Alsina 1986; Aczel 1987) An operator is separable in terms of a unique monotone increasing g ( for all i) and satisfies unanimity and reciprocity if and only if is of the form

(3)

with an arbitrary odd function. Here, exp is the exponential function, and exp a corresponds to e^a.

Let us now consider another condition: positive homogeneity. An aggregation operator satisfies positive homogeneity when the following equation holds.

(4)

for

If we add positive homogeneity to the conditions above we have the following result.

Proposition 2.3

(Aczel and Alsina 1986; Aczel 1987) An operator is separable in terms of a unique monotone increasing g and satisfies unanimity, positive homogeneity (Equation (4)), and reciprocity (Equation (2)) if and only if is the geometric mean:

The AHP Process for the Weighted Mean

This section focuses on the derivation of priorities, or weight learning, for the weighted mean used in the AHP (Saaty 1980; Saaty 1987).

The approach described here is also valid for other quasi-arithmetic means, the OWA operator and its family, as well as the WOWA operator. We discuss this briefly in the next section.

Following the notation used previously, we consider that an aggregation operator aggregates data from a set of n information sources. Let be the set of information sources or variables to be aggregated.

Within the AHP framework, the weights (w₁, …,w_n) for sources or variables X₁, …, X_n are obtained as follows.

• Step 1. The expert is interviewed and is asked about the importance of source X_i with respect to source X_j for all pairs of sources i, j. In the original formulation, this comparison was given in a linguistic scale, with a numerical translation. Because of that, the outcome of this step is a matrix of numerical values.

• Step 2. The consistency of the matrix is studied. We say that a matrix is consistent when the following holds:

  • for all i, j, k, and

  • for all i, j.

  If the matrix is consistent, we can define

  using any row s. It is easy to prove that the weights defined in this way do not depend on the selected row s, and that . In this case, the process is finished.

• Step 3. If the matrix is not consistent, find weights w_i such that w_i/w_j approximate a_ij.

Several approaches exist for the case in which the matrix is not consistent. Golany and Kress (1993) and Ishizaka and Lusti (2006) classify methods into two groups: the eigenvalue approach and the methods minimizing the distance between the user-defined matrix and the nearest consistent matrix.

Crawford and Williams (1985) proposed to select the weighting vector w that minimizes the following difference:

They prove (Theorem 3 in Crawford and Williams 1985) that the solution of this minimization problem when matrices are not consistent, but they are such that and for all i, j, is the weighting vector (w₁, …,w_n) where w_k is defined as follows:

(5)

This way to derive the weights is known as the logarithmic least square method and also as the geometric mean because Equation (5) is the geometric mean. In the next section we consider this problem when, instead of a single value a_ij we have a set of values.

It is easy to prove that if for all i, j, then . Note that

To have weights that add to one, as required by the weighted mean, w_k are normalized . It is clear that if approximate a_ij, the same applies to because

The approach described here was defined by the weighted mean, but it can be applied to any quasi-arithmetic mean because they also require a weighting vector, and the interpretation of this vector is the same as that in the weighted mean. It is similar for the OWA operator. In this case, weights correspond to the importance of the values instead of information sources, but the approach described here applies if the expert is interviewed about the importance of the ith value with respect to the jth value. The same applies to OWA-like operators and the WOWA operator, which uses two types of weights. See Torra and Narukawa (2007) for further discussion.

THE GEOMETRIC MEAN APPROACH FOR SET-VALUED MATRICES

Let us consider now the problem of having a set-valued matrix. That is, we have a matrix , in which in each position a_ij, instead of having a single value, we have a set of values. That is, for each pair of variables (x_i, x_j), we have that the relative preference of x_i over x_j is a set of values. In the more general case, we denote this set by

A structure of this form is the HMPR in, for example, Xia and Xu (2013), Zhang and Wu (2014), Xu (2014). They require values in the [1/9,9] interval, that C(i, j) and C(j, i) have the same number of elements, and that the elements in C(i, j) and in C(j, i) can be matched so that the multiplication of any matched pair is one. We do not require here, a priori, any of these conditions. We will discuss some of these issues later.

The number of elements in C(i, j) can be different for different values of i and j, nevertheless, for the sake of simplicity, when no confusion arises we will use k without subindexes. That is, . We call such a matrix an structure and denote it by .

Inspired by the approach in Crawford and Williams (1985), given the structure , we consider the derivation of priorities (elicitation of weights) w_i as follows.

Problem 3.1

Derivation of priorities. Given an structure , find weights that minimize the following distance,

where is a separable aggregation operator that satisfies idempotency and reciprocity.

The conditions on are based on the following considerations.

1. We consider idempotency for simplicity and technical reasons. Note that idempotency is necessary only when we have a multiset instead of a set.

2. Reciprocity is necessary when we expect that sets

   and
   are such that (i) they have the same number of elements (i.e., ), and (ii) for each , there is an element such that , and for each there is an element such that ; in addition we have that the following equality holds

   Note that this equation is related to the consistency of a_ij and a_ji (i.e., ).

The solution of this problem is given in the next theorem. In the theorem a vector is a weighting vector when and . When weights are required to satisfy they can be normalized as in the AHP. Normalization is needed so that the aggregation operator satisfies unanimity.

Proposition 3.2

(Torra 2014) Let M be an structure , where for each cell (i, j) we have the set , let be a given separable aggregation function satisfying idempotency and reciprocity; then, the weighting vector (w₁, …, w_n) (i.e., and ) that is a solution of the minimization problem

is given by weights

(6)

for .

In addition, is of the form

(7)

where is an arbitrary odd function.

Proof

A detailed proof of this proposition can be found in Torra (2014). The proof is in two parts. In the first part we prove that from Proposition 2.2 it follows Expression 7 for the given operator . In the second part, the expression for the weights is obtained. This second part starts deriving and making the corresponding expression equal to zero. In this way we obtain

(8)

Using , or , this equation leads to Equation (6).

The proof finishes checking that the weights satisfy .□

The solution presented here characterizes a way to derivate priorities. The following function is considered in Zhang and Wu (2014) heuristically:

Note that this expression is a particular case of Expression (7). So our result justifies this heuristic selection, although other aggregation operators are also possible.

Note that the constraint that satisfies reciprocity does not have any role in the proof and any function would be equally acceptable. If we use a function that does not satisfy reciprocity, then we have that Equation (6) is still the solution. However, we will have ratio weights that approximate a matrix , which does not necessarily satisfy reciprocity even when the data in the {C(i, j)}_ij structure satify it.

We illustrate the previous proposition with an example.

Example 3.3

Let , which, as shown in Proposition 2.3, satisfies unanimity and reciprocity (and positive homogeneity). Let us consider the following structure:

For this data, the aggregated matrix is

The solution using Proposition 3.2 is the following vector:

We can observe that the product of these weights is one, so the constraint is satisfied.

Note that the application of Equation (5) to the aggregated matrix is not appropriate because the aggregated matrix is not consistent.

On the Consistency of the Set-Valued Matrices

In our results, we have focused on the properties of the aggregation operator so that the outcome of the method for deriving priorities led to an approximation of the aggregated matrix that is consistent. Nevertheless, we have not discussed the consistency of the set-valued matrix. Note also that Proposition 3.2 does not make any assumption on the values in the set-valued matrix. This is a difference with respect to the results for hesitant multiplicative preference relations (as found in Xia and Xu 2013) in which some conditions are required in the set-valued matrix (reciprocity and ).

Example 3.3 illustrates this fact; we do not make any requirement for the values in a cell C(i, j) with respect to those in the cell C(j, i). Note that we have , but . Similarly, no requirements are given for the values in the cells C(i, i) for i. Nevertheless, it can be seen in Equation (6) that the computation of the weights does not use the values of this set, and the solution always approximates C(i, i) by 1 as .

Following the approach in AHP, in which the original matrix is not required to satisfy the consistency equations but the solution does, the conditions of structures relax the conditions of HMPR and require only the following:

• C1. for all i.

This condition is clearly assumed by any interviewed expert.

Now we consider the definition of a function reconcile that, given a structure, returns another one with extra constraints. In particular, the new structure shall satisfy those of HMPR (e.g., ), and the multiplicative relationship . To do this, the function adds first the reciprocals and then the values needed for the multiplicative relationship.

We propose two variations of this function. We will discuss their properties. The first definition of the function is as follows.

Algorithm 1. Reconcile

Input. structure M defined by C(i, j)_i,j

Output. structure

• Step 1.

• Step 2.

• Step 3. Return

Step 1 adds the reciprocals and Step 2 adds elements to the structure so that the multiplicative relationship is satisfied for the elements in C₀. The idea behind this algorithm is that each element in C(i, k) and each element in C(k, j) should have its product in the structure.

One may consider whether this algorithm returns a consistent structure. That is, whether when and , it also holds . Unfortunately, this is false. So, one step does not lead to a consistent structure.

In addition, successive application of this process does not converge. That is, if we denote by r (from reconcile) this algorithm, r(M) is the application of this algorithm to a given structure, and r²(M) is r(r(M)) (i.e., the application of this algorithm to the outcome of the application of this algorithm to the structure M), and rⁿ(M) is for , we have that we cannot ensure that for any structure M there is n with . In short, multiple applications of this algorithm do not lead to a consistent structure either.

Let us show this with an example. Table 1 illustrates it. We consider just three cells in the structure. They are, respectively, positions (1,2), (2,3) and (1,3). Initially, there is a single value in each of these cells of the structure. These values are, respectively, , and . The table illustrates the values added in successive applications of the algorithm. The initial value is represented in the table in column C₀, the first addition in column C₁, the second in column C₂, and the third application in column C₃. We consider only one addition per row in each step and the effects in other rows.

TABLE 1 Example of the Application of the Algorithm

	C0	C1	C2	C3
Ci(1, 2)
Ci(2, 3)
Ci(1, 3)

It is easy to see that, in subsequent steps, we will need to add new terms, for example and for .

An alternative to this approach is to check for each a_ij if there is any value k for which there are a_ik and a_kj for which . If this is not the case, then we add a value so that the equation is satisfied. This approach is used in the following algorithm. In this case we can prove that the algorithm leads to a structure that satisfies the multiplicative constraint:

Algorithm 2. Reconcile

Input. structure M defined by C(i, j)_i,j

Output. structure

• Step 1.

• Step 2. For all i, j such that

  • Step 2.1. exists k such that ?

  • Step 2.2. if false(mult(i, j)) then {

    for some k (the selection of k is arbitrary)

    }

• Step 3. Return

This algorithm adds to cell C(k, j). A similar algorithm can be defined adding to cell C(i, k).

The application of this algorithm converges. That is, if r′ is this algorithm then . The following proposition proves this fact.

Proposition 3.4

When we add a_kj in Step 2.2 in the Algorithm 2 Reconcile, there is a k′ for which and .

Proof

If we add a_kj to C₀(k, j), it means that there is the value and . Then, because of Step 1 we have also . Let a_k,i denote this value .

So, for a_kj we have and a_ij such that . Therefore, the proposition is proven.□

This algorithm, using the notation of Xia and Xu (2013) returns an HMPR that satisfies the multiplicative constraint.

• C2. For all i, j if , then there is .

The multiplicative constraint is defined formally as follows:

• C3. For all i, j if , then there exists k such that and and .

Definition 3.5

Let M be an structure or a hesitant multiplicative preference relation. We say that M is a consistent hesitant multiplicative preference relation (cHMPR) if it satisfies constraints C1, C2, and C3.

Note that the function reconcile can be applied to an structure but also to a standard real-valued matrix.

This function reconcile can be seen as a function consistentize, That is, as a function that makes the matrix consistent. From this point of view, the function permits us to revisit the AHP because now, given any inconsistent (standard real-valued) matrix in the sense of Saaty (1980) we can (i) just proceed to derivate the priorities from the matrix using standard procedures or (ii) consistentize the matrix and then derivate the weights from the consistent structure using Proposition 3.2. Note that the structure is no longer a matrix because now it is set-valued.

CONCLUSIONS

In this study we have considered the problem of finding the optimal weights corresponding to a set-based matrix.

As future work we will consider the following two problems. First, in Proposition 3.2, we consider that there is a single aggregation function for all the cells of the table. A more general situation would be when we do not require from the start all aggregation operators to be the same.

Another line of work is to study the solution of our optimization problem to the reconciled data and its similarity to the solution of the approach in Crawford and Williams (1985) to the original data. We will also compare the aggregated matrix of a consistent structure with the one that the AHP vectors permit us to build.

FUNDING

This work is partially funded by TIN2011-27076-C03-03 of the Spanish Government.

Additional information

Funding

This work is partially funded by TIN2011-27076-C03-03 of the Spanish Government.