Geometric Least Square Models for Deriving [0,1]-Valued Interval Weights from Interval Fuzzy Preference Relations Based on Multiplicative Transitivity

Abstract

This paper presents a geometric least square framework for deriving [0,1]-valued interval weights from interval fuzzy preference relations. By analyzing the relationship among [0,1]-valued interval weights, multiplicatively consistent interval judgments, and planes, a geometric least square model is developed to derive a normalized [0,1]-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.

Xuan Yang 1, 2 and Zhou-Jing Wang 3

Academic Editor:Roman Lewandowski

1, School of Accounting, Zhongnan University of Economics and Law, Wuhan, Hubei 430073, China
2, School of Accounting, Zhejiang University of Finance & Economics, Hangzhou, Zhejiang 310018, China
3, School of Information, Zhejiang University of Finance & Economics, Hangzhou, Zhejiang 310018, China

Received 11 November 2014; Revised 6 June 2015; Accepted 8 June 2015; 13 July 2015

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.

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]) [figure omitted; refer to PDF] whose element [figure omitted; refer to PDF] denotes the fuzzy preference degree of the object [figure omitted; refer to PDF] over [figure omitted; refer to PDF] and satisfies [figure omitted; refer to PDF] , and the additive reciprocal property of [figure omitted; refer to PDF] . 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 [figure omitted; refer to PDF] -valued interval weights (or called interval probabilities [8-10]). Different methods have been developed to derive [figure omitted; refer to PDF] -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 [figure omitted; refer to PDF] -valued interval weights. Based on Xu and Chen's multiplicative consistency, Genç et al. [12] propose a formula to determine a [figure omitted; refer to PDF] -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 [figure omitted; refer to PDF] -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 [figure omitted; refer to PDF] -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 [figure omitted; refer to PDF] -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 [figure omitted; refer to PDF] -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 [figure omitted; refer to PDF] -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 [figure omitted; refer to PDF] -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 [figure omitted; refer to PDF] -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 [figure omitted; refer to PDF] be an alternative set and let [figure omitted; refer to PDF] be a pairwise comparison matrix on [figure omitted; refer to PDF] ; if [figure omitted; refer to PDF] satisfies [figure omitted; refer to PDF] then [figure omitted; refer to PDF] is called a fuzzy preference relation.

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

Definition 1 (see [16]).

Let [figure omitted; refer to PDF] be a fuzzy preference relation with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . [figure omitted; refer to PDF] is said to have multiplicative consistency, if it satisfies transitivity condition: [figure omitted; refer to PDF]

It is obvious that (2) is equivalent to any of the following equations: [figure omitted; refer to PDF]

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 [figure omitted; refer to PDF] on [figure omitted; refer to PDF] is characterized by an interval-valued pairwise comparison matrix [figure omitted; refer to PDF] satisfying the following condition: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] denotes an interval importance or preference degree of [figure omitted; refer to PDF] over [figure omitted; refer to PDF] .

Given two interval numbers [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , their arithmetic operation laws are summarized as follows:

(1) Addition: [figure omitted; refer to PDF] .

(2) Subtraction: [figure omitted; refer to PDF] .

(3) Multiplication: [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] .

(4) Division: [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] .

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

Definition 3 (see [15]).

Let [figure omitted; refer to PDF] be an IFPR with [figure omitted; refer to PDF] . [figure omitted; refer to PDF] is said to have multiplicative consistency, if [figure omitted; refer to PDF] satisfies [figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] be the priority weight of the alternative [figure omitted; refer to PDF] for [figure omitted; refer to PDF] ; then the ratio-based interval preference intensity of the alternative [figure omitted; refer to PDF] over [figure omitted; refer to PDF] can be determined as [figure omitted; refer to PDF] . As per interval arithmetic, one can obtain [figure omitted; refer to PDF] . 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, [figure omitted; refer to PDF] -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 [figure omitted; refer to PDF] and [figure omitted; refer to PDF] -valued judgment [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] , implying that [figure omitted; refer to PDF] is the [figure omitted; refer to PDF] -valued interval preference intensity of the alternative [figure omitted; refer to PDF] over [figure omitted; refer to PDF] . Based on this idea, the following transformation formula is proposed by Wang and Li [15] to convert [figure omitted; refer to PDF] -valued interval weight vector [figure omitted; refer to PDF] into multiplicatively consistent IFPR [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is a normalized [figure omitted; refer to PDF] -valued interval weight vector such that [figure omitted; refer to PDF]

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

3. A Geometric Least Square Model for an IFPR

This section develops a geometric least square model to derive normalized [figure omitted; refer to PDF] -valued interval weights from an IFPR.

As per (6), for IFPR [figure omitted; refer to PDF] with [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ), if there exists normalized [figure omitted; refer to PDF] -valued interval weight vector [figure omitted; refer to PDF] satisfying [figure omitted; refer to PDF] then [figure omitted; refer to PDF] has multiplicative consistency.

As [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , it follows from (9) that [figure omitted; refer to PDF]

Clearly, (10) can be equivalently converted into [figure omitted; refer to PDF]

According to the theory of analytical geometry, we can view [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) as a number of planes. Thus, the corresponding normalized [figure omitted; refer to PDF] -valued interval weight vector [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) have no unified intersection point. In this case, one has to seek [figure omitted; refer to PDF] -space point [figure omitted; refer to PDF] satisfying (7) as close to each plane as possible.

Let [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] denote the distances from the point [figure omitted; refer to PDF] to the planes [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively.

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

Since [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , one can obtain [figure omitted; refer to PDF] Therefore, solutions to model (13) are found by solving the following optimization model: [figure omitted; refer to PDF]

Solving (15), one gets a normalized [figure omitted; refer to PDF] -valued interval weight vector expressed as [figure omitted; refer to PDF] .

Substituting [figure omitted; refer to PDF] into (6), we obtain a multiplicatively consistent IFPR as [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

If [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is the optimal objective value of (15), then all of the DM's pairwise judgments in [figure omitted; refer to PDF] are expressed as (17). It follows that [figure omitted; refer to PDF] is the same as [figure omitted; refer to PDF] . Thus, [figure omitted; refer to PDF] is a multiplicatively consistent IFPR.

Example 4.

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

Solving model (15) by the Optimization Modelling Software Lingo 11, one can obtain the following optimal [figure omitted; refer to PDF] -valued interval weight vector: [figure omitted; refer to PDF]

By (8), the matrix of the possibility degree is determined as [figure omitted; refer to PDF] Summing all of elements in each line of [figure omitted; refer to PDF] , we obtain [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . As [figure omitted; refer to PDF] , the four alternatives are ranked as [figure omitted; refer to PDF] .

By (17), the corresponding multiplicatively consistent IFPR is determined as [figure omitted; refer to PDF]

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 [figure omitted; refer to PDF] to derive priority weights that are summarized in Table 1.

Table 1: A comparative study for the IFPR [figure omitted; refer to PDF] .

Model	Reference	Priority weight vector [figure omitted; refer to PDF]	Ranking
(M-8) and (M-9)	Xu and Chen [11]	[figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Equations (22) and (23)	Genç et al. [12]	[figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Algorithm [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF]	Lan et al. [13]	[figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF]	[figure omitted; refer to PDF]
(MOD5) [figure omitted; refer to PDF]	Xia and Xu [14]	[figure omitted; refer to PDF] , [figure omitted; refer to PDF] [figure omitted; refer to PDF] , [figure omitted; refer to PDF] [figure omitted; refer to PDF] , [figure omitted; refer to PDF] [figure omitted; refer to PDF] , [figure omitted; refer to PDF]	[figure omitted; refer to PDF] , [figure omitted; refer to PDF] [figure omitted; refer to PDF] , [figure omitted; refer to PDF]
Equation (15)	This paper	[figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF]	[figure omitted; refer to PDF]

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 [figure omitted; refer to PDF] -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, [figure omitted; refer to PDF] is judged to be a consistent IFPR. One can verify that [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , 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 [figure omitted; refer to PDF] is furnished by the DM [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) to express his/her preferences on the alternative set [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] be a set of [figure omitted; refer to PDF] DMs, and let [figure omitted; refer to PDF] be an importance weight vector of [figure omitted; refer to PDF] DMs or a relative weight vector of IFPRs [figure omitted; refer to PDF] , which is known and satisfies [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] .

As different DMs generally have different subjective preferences or pair-wise judgments, it is nearly impossible to seek a common intersection point for planes [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . In order to generate a unified [figure omitted; refer to PDF] -valued interval weight vector for all individual IFPRs, the distances from a point to the planes are introduced as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] denote the distances from the [figure omitted; refer to PDF] -space point [figure omitted; refer to PDF] to the planes [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively.

Once again, the smaller the sum of the values of the distances [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , the better the IFPR [figure omitted; refer to PDF] from the viewpoint of the multiplicative consistency. As different IFPRs [figure omitted; refer to PDF] have different importance weights, a reasonable priority weight vector will be obtained by minimizing the weighted sum of these distances. Therefore, the following geometric least square model is established to derive a group [figure omitted; refer to PDF] -valued interval weight vector directly from individual IFPRs: [figure omitted; refer to PDF]

As [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] Thus, solutions to model (23) are determined by solving the following geometric least square model: [figure omitted; refer to PDF]

Solving this model, a group [figure omitted; refer to PDF] -valued interval weight vector is determined as [figure omitted; refer to PDF]

4.2. Determination of Importance Weights of Individual IFPRs

Models (23) and (25) are developed by assuming the importance weights of DMs (or experts) or the relative weights of [figure omitted; refer to PDF] IFPRs to be known. However, in many real-world situations, it is difficult to directly assign importance weights to DMs or IFPRs because their importance depends on many factors such as expert's assessment level, DM's knowledge, and expertise related to the decision problem domain. In other words, the importance weights of DMs or the relative weights of [figure omitted; refer to PDF] IFPRs will have to be determined.

In group decision analysis, if [figure omitted; refer to PDF] IFPRs are the same, it is logical to assign their importance the same weights; that is, [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] . In this case, model (25) is reduced to (15). If the IFPR [figure omitted; refer to PDF] is much different from the others, its importance weight should be small and the geometric mean of the difference ratios between [figure omitted; refer to PDF] and the others is large. Conversely, if [figure omitted; refer to PDF] is very similar to the others, its importance should be high and the geometric mean of the difference ratios between [figure omitted; refer to PDF] and the others is small. In order to determine the relative weights of individual IFPRs, a geometric mean based difference ratio between any two IFPRs is introduced as follows.

Definition 5.

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be any two IFPRs; then, the difference ratio between [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is defined as [figure omitted; refer to PDF]

Obviously, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] , one has [figure omitted; refer to PDF] . The larger the difference ratio [figure omitted; refer to PDF] is, the less similar the [figure omitted; refer to PDF] is to [figure omitted; refer to PDF] .

In order to derive the relative weights of individual IFPRs, the geometric average difference ratio between one IFPR and the others is introduced as follows.

Definition 6.

Let [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) be individual IFPRs; then, the geometric average difference ratio between [figure omitted; refer to PDF] and the others is defined as [figure omitted; refer to PDF]

It is obvious that [figure omitted; refer to PDF] . The smaller the [figure omitted; refer to PDF] , the more important the [figure omitted; refer to PDF] among individual IFPRs. In particular, if [figure omitted; refer to PDF] , one can obtain [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] IFPRs are completely the same. Therefore, the following formula can be employed to determine the relative weight of [figure omitted; refer to PDF] : [figure omitted; refer to PDF]

Clearly, we have [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] .

Based on the above analyses, we now develop an approach for deriving a group [figure omitted; refer to PDF] -valued interval weight vector directly from individual IFPRs with unknown importance weights. The approach is described in the following steps.

Step 1.

For a group decision making problem with an alternative set [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] be the set of [figure omitted; refer to PDF] DMs. The DMs [figure omitted; refer to PDF] furnish their preferences by means of IFPRs [figure omitted; refer to PDF] , where the relative importance weights of [figure omitted; refer to PDF] are unknown, and [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] .

Step 2.

Calculate the difference ratios between [figure omitted; refer to PDF] and [figure omitted; refer to PDF] as per (27).

Step 3.

Utilize (28) to calculate the difference ratio between [figure omitted; refer to PDF] and the others for all [figure omitted; refer to PDF] .

Step 4.

Utilize (29) to obtain the relative weight of [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] .

Step 5.

Determine a group [figure omitted; refer to PDF] -valued interval weight vector [figure omitted; refer to PDF] by solving model (25).

Step 6.

Calculate the possibility degree [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) as per (8).

Step 7.

Construct the possibility degree matrix [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] .

Step 8.

Adding all values in each row of [figure omitted; refer to PDF] , we obtain [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ).

Step 9.

As per the decreasing order of [figure omitted; refer to PDF] , a ranking order for all decision alternatives is derived, and " [figure omitted; refer to PDF] being preferred to [figure omitted; refer to PDF] " is expressed as [figure omitted; refer to PDF] .

5. An Application to the Enterprise Resource Planning Software Product Selection Problem

This section applies the proposed approach in Section 4 to an enterprise resource planning (ERP) software product selection problem that concerns group decision making with a hierarchical structure.

The ERP system has an important impact on improving the productivity of the organizations. However, the implementation of an ERP system is often very expensive and complex. Therefore, selecting the best suitable ERP software product is a vital decision making problem of the organizations when they aim to buy a ready ERP system in the market. Many factors or criteria impact the ERP software product selection [17]. In this case study, the ERP software product selection is made by the following five critical evaluation criteria: functionality ( [figure omitted; refer to PDF] ), cost and customization ( [figure omitted; refer to PDF] ), reliability ( [figure omitted; refer to PDF] ), compatibility ( [figure omitted; refer to PDF] ), and market position and reputation ( [figure omitted; refer to PDF] ).

Although there are many potential ERP software products in the market, only five of them, denoted by [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , are identified as candidates. A committee consisting of three experts ( [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ) is set up to evaluate the five ERP software products, and its objective is to select the best one based on the above criterion scheme. The hierarchy of this ERP software product selection problem is shown in Figure 1.

Figure 1: Decision hierarchy of an ERP software product selection problem.

[figure omitted; refer to PDF]

As the importance weights of the five criteria are to be determined, each expert [figure omitted; refer to PDF] compares each pair of the criteria and provides his/her judgments by means of an IFPR [figure omitted; refer to PDF] . Consider [figure omitted; refer to PDF]

On the other hand, the importance of the three experts is also unknown. Therefore, we need firstly to determine the relative weights of [figure omitted; refer to PDF] .

By (27), one can obtain the difference ratios between [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) as follows: [figure omitted; refer to PDF]

As per (28), the difference ratios [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) are calculated as [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

According to (29), the relative weights of [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) are obtained as [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

Next, substituting [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) into (25) and solving this model, one gets normalized criterion weight vector [figure omitted; refer to PDF] [figure omitted; refer to PDF] .

Based on the criterion scheme, each expert [figure omitted; refer to PDF] compares each pair of the five ERP software products with respect to each criterion [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) and furnishes his/her judgments by an IFPR [figure omitted; refer to PDF] . Consider [figure omitted; refer to PDF]

For each criterion [figure omitted; refer to PDF] , by using (27), (28), and (29), the difference ratios [figure omitted; refer to PDF] and the relative weights of [figure omitted; refer to PDF] are determined as shown in Table 2.

Table 2: Difference ratios and relative weights for individual IFPRs.

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
GMDR	Weight	GMDR	Weight	GMDR	Weight	GMDR	Weight	GMDR	Weight
1	1.3051	0.3961	1.2542	0.3575	1.3571	0.3004	1.2362	0.3560	1	1/3
2	1.4341	0.3605	1.2542	0.3575	1.1649	0.3498	1.5281	0.2880	1	1/3
3	2.1244	0.2434	1.5729	0.2850	1.1649	0.3498	1.2362	0.3560	1	1/3

For each criterion [figure omitted; refer to PDF] , plugging [figure omitted; refer to PDF] and their relative weights into (25) and solving this model, one can obtain a normalized [figure omitted; refer to PDF] -valued interval weight vector for [figure omitted; refer to PDF] with respect to the criterion [figure omitted; refer to PDF] , denoted by [figure omitted; refer to PDF] , as listed in columns 1-5 in Table 3, where the first row gives the criterion weights [figure omitted; refer to PDF] derived earlier.

Table 3: Local [figure omitted; refer to PDF] -valued interval weights and the aggregated interval weights.

Candidate	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	Aggregated interval weights
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

Similar to the treatment in Wang and Li [15], the following linear programs given by Bryson and Mobolurin [18] are applied to aggregate local [figure omitted; refer to PDF] -valued interval weights into the aggregated interval weights for [figure omitted; refer to PDF] . Consider [figure omitted; refer to PDF] where [figure omitted; refer to PDF] (for [figure omitted; refer to PDF] ) are decision variables.

In (33), [figure omitted; refer to PDF] (for [figure omitted; refer to PDF] ) are the normalized [figure omitted; refer to PDF] -valued interval weights of the five criteria, and [figure omitted; refer to PDF] (for [figure omitted; refer to PDF] ) are the normalized [figure omitted; refer to PDF] -valued interval weights for the five alternatives over the criterion [figure omitted; refer to PDF] . They are determined and shown in Table 3.

Solving (33) yields the aggregated [figure omitted; refer to PDF] -valued interval weight [figure omitted; refer to PDF] for [figure omitted; refer to PDF] as listed in the last column in Table 3.

By (8), we obtain the possibility degree matrix as follows: [figure omitted; refer to PDF]

Adding up all of elements in each row of [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . As [figure omitted; refer to PDF] , the five ERP software products are ranked as [figure omitted; refer to PDF] .

6. Conclusions

Derivation of priority weights from IFPRs plays an important role for MCDM with interval fuzzy preference information. In this paper, we have analyzed the relationship among the normalized [figure omitted; refer to PDF] -valued interval weights, multiplicatively consistent interval judgments, and planes. A geometric least square model has been developed for deriving [figure omitted; refer to PDF] -valued interval weights from any IFPR and extended to generate a group [figure omitted; refer to PDF] -valued interval weight vector directly from individual IFPRs, whose relative weights are assumed to be known. We have introduced the notion of the geometric average difference ratio between one IFPR and the others and applied it to determine the relative importance weights of individual IFPRs. A geometric least squares based approach has been put forward for group decision making with IFPRs. We have provided a numerical example and comparative analyses to illustrate the validity of the proposed models and presented a case study to show that the proposed framework is operational in practice.

In the future, we will focus on the ratio-based geometric similarity measure on IFPRs and its application to consensus models of group decision making.

Acknowledgments

The research is supported by the Zhejiang Provincial Natural Science Foundation of China under Grant LY15G010004 and the National Natural Science Foundation of China under Grant 71271188.

Conflict of Interests

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

References

[1] T. L. Saaty The Analytic Hierarchy Process , McGraw-Hill, New York, NY, USA, 1980.

[2] F. Chiclana, E. Herrera-Viedma, F. Alonso, S. Herrera, "Cardinal consistency of reciprocal preference relations: a characterization of multiplicative transitivity," IEEE Transactions on Fuzzy Systems , vol. 17, no. 1, pp. 14-23, 2009.

[3] M. Xia, Z. Xu, J. Chen, "Algorithms for improving consistency or consensus of reciprocal [0,1]-valued preference relations," Fuzzy Sets and Systems , vol. 216, pp. 108-133, 2013.

[4] X. Liu, Y. Pan, Y. Xu, S. Yu, "Least square completion and inconsistency repair methods for additively consistent fuzzy preference relations," Fuzzy Sets and Systems , vol. 198, pp. 1-19, 2012.

[5] R.-H. Chiu, L.-H. Lin, S.-C. Ting, "Evaluation of green port factors and performance: a fuzzy AHP analysis," Mathematical Problems in Engineering , vol. 2014, 2014.

[6] C.-C. Chou, K.-W. Yu, "Application of a new hybrid fuzzy AHP model to the location choice," Mathematical Problems in Engineering , vol. 2013, 2013.

[7] Z. Xu, "On compatibility of interval fuzzy preference relations," Fuzzy Optimization and Decision Making , vol. 3, no. 3, pp. 217-225, 2004.

[8] T. Entani, K. Sugihara, "Uncertainty index based interval assignment by interval AHP," European Journal of Operational Research , vol. 219, no. 2, pp. 379-385, 2012.

[9] P. Guo, Y. Wang, "Eliciting dual interval probabilities from interval comparison matrices," Information Sciences , vol. 190, pp. 17-26, 2012.

[10] K. Sugihara, H. Ishii, H. Tanaka, "Interval priorities in AHP by interval regression analysis," European Journal of Operational Research , vol. 158, no. 3, pp. 745-754, 2004.

[11] Z. Xu, J. Chen, "Some models for deriving the priority weights from interval fuzzy preference relations," European Journal of Operational Research , vol. 184, no. 1, pp. 266-280, 2008.

[12] S. Genç, F. E. Boran, D. Akay, Z. Xu, "Interval multiplicative transitivity for consistency, missing values and priority weights of interval fuzzy preference relations," Information Sciences , vol. 180, no. 24, pp. 4877-4891, 2010.

[13] J. Lan, M. Hu, X. Ye, S. Sun, "Deriving interval weights from an interval multiplicative consistent fuzzy preference relation," Knowledge-Based Systems , vol. 26, pp. 128-134, 2012.

[14] M. Xia, Z. Xu, "Interval weight generation approaches for reciprocal relations," Applied Mathematical Modelling , vol. 38, no. 3, pp. 828-838, 2014.

[15] Z.-J. Wang, K. W. Li, "Goal programming approaches to deriving interval weights based on interval fuzzy preference relations," Information Sciences , vol. 193, pp. 180-198, 2012.

[16] T. Tanino, "Fuzzy preference orderings in group decision making," Fuzzy Sets and Systems , vol. 12, no. 2, pp. 117-131, 1984.

[17] T. Gürbüz, S. E. Alptekin, G. I. Alptekin, "A hybrid MCDM methodology for ERP selection problem with interacting criteria," Decision Support Systems , vol. 54, no. 1, pp. 206-214, 2012.

[18] N. Bryson, A. Mobolurin, "An action learning evaluation procedure for multiple criteria decision making problems," European Journal of Operational Research , vol. 96, no. 2, pp. 379-386, 1997.