Abstract

This paper presents a geometric least square framework for deriving -valued interval weights from interval fuzzy preference relations. By analyzing the relationship among -valued interval weights, multiplicatively consistent interval judgments, and planes, a geometric least square model is developed to derive a normalized -valued interval weight vector from an interval fuzzy preference relation. Based on the difference ratio between two interval fuzzy preference relations, a geometric average difference ratio between one interval fuzzy preference relation and the others is defined and employed to determine the relative importance weights for individual interval fuzzy preference relations. A geometric least square based approach is further put forward for solving group decision making problems. An individual decision numerical example and a group decision making problem with the selection of enterprise resource planning software products are furnished to illustrate the effectiveness and applicability of the proposed models.

1. Introduction

The preference relation is a common framework for expressing decision-makers’ (DMs’) pairwise comparison results in multicriteria decision making (MCDM). One widely used preference relation takes the form of the multiplicative preference relation, which was introduced by Saaty [1] to structure DMs’ pairwise comparison ratios in the analytic hierarchy process (AHP). Another popularly used preference relation takes the form of a fuzzy preference relation (also called a reciprocal preference relation [2, 3]) whose element denotes the fuzzy preference degree of the object over and satisfies , and the additive reciprocal property of . Over the last three decades, fuzzy preference relations have been extensively studied [4] and the fuzzy AHP has been widely applied to various MCDM problems such as the green port evaluation [5] and the location selection [6], to name a few.

All of judgments in a fuzzy preference relation are characterized by crisp values. However, in many real-world situations, DMs’ subjective judgments may be bounded between lower and upper bounds due to complexity and indeterminacy of decision problems. Therefore, the concept of interval fuzzy preference relations (IFPRs) is introduced by Xu [7] to describe imprecise and uncertain judgment information, and an increasing research interest has been concentrating on employing IFPRs to help DMs make their decision analyses.

An important research topic for MCDM with preference information is to derive priority weight vectors from preference relations. As preference information contains two kinds of uncertainty, that is, DM’s judgments and inconsistency among comparisons, the derived priority weights should be -valued interval weights (or called interval probabilities [8–10]). Different methods have been developed to derive -valued interval weights from IFPRs. Xu and Chen [11] define additively consistent IFPRs and multiplicatively consistent IFPRs from the viewpoint of the feasible regions and develop two linear-programming-based approaches for obtaining -valued interval weights. Based on Xu and Chen’s multiplicative consistency, Genç et al. [12] propose a formula to determine a -valued interval weight vector of an IFPR, in which the original IFPR is converted into the one with multiplicative consistency. They also show that the derived -valued interval weight vector is the same as the result obtained by the approach given in Xu and Chen [11]. Lan et al. [13] put forward an exchange method between additively consistent IFPRs and multiplicatively consistent IFPRs and devise a parametric algorithm to obtain -valued interval weights by converting a multiplicatively consistent IFPR into an additively consistent IFPR. Xia and Xu [14] establish two parametric programming models to generate -valued interval weights of an IFPR. From the viewpoint of interval arithmetic, Wang and Li [15] define additively consistent IFPRs, multiplicatively consistent IFPRs, and weakly transitive IFPRs and design two goal programs to generate -valued interval weights for individual and collective decisions.

The literature review indicates that among the priority methods mentioned above for IFPRs, most of them are developed according to the feasible-region-based consistency definitions and are only applicable to one IFPR. Although Wang and Li’s approach [15] may be used to derive a group -valued interval weight vector directly from individual IFPRs, it requires the importance weights of DMs or the relative weights of individual IFPRs to be known. It is very hard to assign the subjective weights to DMs in some group decision situations, such as the group decision making problem with a hierarchical structure in Section 5. On the other hand, so far little research has been found on employing the idea of geometric least squares to generate priority weights from IFPRs and determining the relative weights of individual IFPRs in group decision situations. In this paper, we develop a geometric least square model to derive -valued interval weights from an IFPR. To measure the relative importance of individual IFPRs, the difference ratio between any two IFPRs is introduced to define the geometric average difference ratio between one IFPR and the others. A geometric least square based approach is further developed for solving group decision making problems with unknown DMs’ weights.

The rest of the paper is set out as follows. Section 2 reviews some basic notions related to fuzzy preference relations and multiplicatively consistent IFPRs. A geometric least square model is established for deriving a -valued interval weight vector from an IFPR in Section 3. Section 4 puts forward a method for determining the relative importance weights of individual IFPRs and develops a geometric least square based approach for deriving a group priority weigh vector directly from individual IFPRs. Section 5 provides a case study on the enterprise resource planning software product selection problem. Section 6 draws the main conclusions.

2. Preliminaries

For an MCDM problem, let be an alternative set and let be a pairwise comparison matrix on ; if satisfies then is called a fuzzy preference relation.

The element in gives a -valued importance or fuzzy preference degree of over . As the additive reciprocal property of , the larger the value of , the stronger the preference ratio of over . If , then and is preferred to with the ratio . If , then and is nonpreferred to with the ratio . In particular, if , then , indicating that and are indifferent.

Definition 1 (see [16]). Let be a fuzzy preference relation with , . is said to have multiplicative consistency, if it satisfies transitivity condition:

It is obvious that (2) is equivalent to any of the following equations:

With increasing complexity and indeterminacy in many decision problems, it is often difficult for a DM to furnish crisp preference degrees. To better model vague and uncertain DM’s judgments, Xu [7] introduces the concept of IFPRs.

Definition 2 (see [7]). An IFPR on is characterized by an interval-valued pairwise comparison matrix satisfying the following condition:where denotes an interval importance or preference degree of over .

Given two interval numbers and , their arithmetic operation laws are summarized as follows:(1)Addition: .(2)Subtraction: .(3)Multiplication: , where .(4)Division: , where .

Based on interval arithmetic, Wang and Li [15] introduce the multiplicative transitivity to define consistent IFPRs.

Definition 3 (see [15]). Let be an IFPR with . is said to have multiplicative consistency, if satisfies

Let be the priority weight of the alternative for ; then the ratio-based interval preference intensity of the alternative over can be determined as . As per interval arithmetic, one can obtain . The ratio-based preference intensities are based on the positive real line with the neutral value 1 denoting the indifference between two alternatives. On the other hand, -valued interval judgments in IFPRs are based on the bipolar unit interval scale having the neutral element 0.5. There exists a relation between ratio-based judgment and -valued judgment ; that is, , implying that is the -valued interval preference intensity of the alternative over . Based on this idea, the following transformation formula is proposed by Wang and Li [15] to convert -valued interval weight vector into multiplicatively consistent IFPR , whereand is a normalized -valued interval weight vector such that

Let and be any two -valued interval weights; then, the following possibility degree formula is defined [11] and employed to compare them:

3. A Geometric Least Square Model for an IFPR

This section develops a geometric least square model to derive normalized -valued interval weights from an IFPR.

As per (6), for IFPR with (), if there exists normalized -valued interval weight vector satisfying then has multiplicative consistency.

As for all , it follows from (9) that

Clearly, (10) can be equivalently converted into

According to the theory of analytical geometry, we can view and () as a number of planes. Thus, the corresponding normalized -valued interval weight vector can be seen as an intersection point of these planes.

On the other hand, (11) holds for multiplicatively consistent IFPRs. In the real-life decision situations, IFPRs furnished by DMs are often inconsistent and may not be denoted by (11). In other words, the planes and () have no unified intersection point. In this case, one has to seek -space point satisfying (7) as close to each plane as possible.

Letwhere and denote the distances from the point to the planes and , respectively.

Obviously, the smaller the sum of the values of the distances and is, the closer the is to a multiplicatively consistent IFPR. Therefore, reasonable point can be determined by solving the following geometric least square optimization model:where the constraints are the normalization conditions of the -valued interval weight vector corresponding to (7), and and are decision variables.

Since and for all , one can obtainTherefore, solutions to model (13) are found by solving the following optimization model:

Solving (15), one gets a normalized -valued interval weight vector expressed as .

Substituting into (6), we obtain a multiplicatively consistent IFPR as where

If , where is the optimal objective value of (15), then all of the DM’s pairwise judgments in are expressed as (17). It follows that is the same as . Thus, is a multiplicatively consistent IFPR.

Example 4. We discuss an MCDM problem concerning four decision alternatives , , , and . Denote the alternative set by . A DM compares each pair of alternatives on and yields the following IFPR, which has been examined by Lan et al. [13]:

Solving model (15) by the Optimization Modelling Software Lingo 11, one can obtain the following optimal -valued interval weight vector:

By (8), the matrix of the possibility degree is determined asSumming all of elements in each line of , we obtain , , , and . As , the four alternatives are ranked as .

By (17), the corresponding multiplicatively consistent IFPR is determined as

Next, four different approaches proposed by Xu and Chen [11], Genç et al. [12], Lan et al. [13], and Xia and Xu [14] are applied to the same IFPR to derive priority weights that are summarized in Table 1.

Table 1 demonstrates that the ranking orders are nearly consistent based on the five different models. However, the values of the possibility degree of the obtained -valued interval weights in this paper differ from the results derived from the other methods, which is due to the fact that the approaches adopt different consistency constraints for IFPRs. The transitivity conditions in [11–14] are all based on the feasible-region method; thus, is judged to be a consistent IFPR. One can verify that is not multiplicatively consistent under Definition 3. On the other hand, Xia and Xu’s method [14] can only generate crisp priority weight vectors and yields distinct rankings under different parameter values for this particular IFPR. Lan et al.’s method [13] has to select appropriate parameters and , which seems difficult and complex.

4. Geometric Least Square Models for Group Decision Making with IFPRs

4.1. Derivation of Interval Weights Based on Individual IFPRs with Known Importance Weights

In the real-world situations, a decision is often made by a group of DMs. Suppose that an individual IFPR is furnished by the DM () to express his/her preferences on the alternative set . Let be a set of DMs, and let be an importance weight vector of DMs or a relative weight vector of IFPRs , which is known and satisfies and for all .