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

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,ageometricleastsquaremodelisdevelopedtoderiveanormalized [0, 1] -valuedintervalweightvectorfromanintervalfuzzy 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.


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 0 ≤   ≤ 1,   = 0.5, and the additive reciprocal property of   +   = 1.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 2 Mathematical Problems in Engineering among comparisons, the derived priority weights should be [0, 1]-valued interval weights (or called interval probabilities [8][9][10]).Different methods have been developed to derive [0, 1]-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 [0, 1]-valued interval weights.Based on Xu and Chen's multiplicative consistency, Genc ¸et al. [12] propose a formula to determine a [0, 1]-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 [0, 1]-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 [0, 1]-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 [0, 1]-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 [0, 1]-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 [0, 1]-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 [0, 1]-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 [0, 1]-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.
The element   in  gives a [0, 1]-valued importance or fuzzy preference degree of   over   .As the additive reciprocal property of   +   = 1, the larger the value of   , the stronger the preference ratio   /  of   over   .If   > 0.5, then   /  > 1 and   is preferred to   with the ratio   /  .If   < 0.5, then   /  < 1 and   is nonpreferred to   with the ratio   /  .In particular, if   = 0.5, then   /  = 1, indicating that   and   are indifferent.
Definition 1 (see [16]).Let  = (  ) × be a fuzzy preference relation with 0 <   < 1, ∀,  = 1, 2, . . ., .  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 𝑎
, their arithmetic operation laws are summarized as follows.

Mathematical Problems in Engineering
Clearly, (10) can be equivalently converted into According to the theory of analytical geometry, we can view )  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 Let where  (1)   and  (2)   denote the distances from the point Obviously, the smaller the sum of the values of the distances  (1)   and  (2)   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 [0, 1]-valued interval weight vector  corresponding to (7), and  −  ( = 1, 2, . . ., ) and  +  ( = 1, 2, . . ., ) are decision variables.
Example 4. We discuss an MCDM problem concerning four decision alternatives  1 ,  2 ,  3 , and  4 .Denote the alternative set by  = { 1 ,  2 ,  3 ,  4 }.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 [0, 1]valued interval weight vector: By (8), the matrix of the possibility degree is determined as Next, four different approaches proposed by Xu and Chen [11], Genc ¸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 [0, 1]-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][12][13][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.

Geometric Least Square Models for Group
Decision Making with IFPRs In order to generate a unified [0, 1]-valued interval weight vector for all individual IFPRs, the distances from a point to the planes are introduced as follows: where  (1)   and  (2)   denote the distances from the 2-space point ( − 1 ,  + 1 ,  − 2 ,  + 2 , . . .,  −  ,  +  )  to the planes  −   +  −  +   −  = 0 and  +   −  −  −   +  = 0, respectively.Once again, the smaller the sum of the values of the distances  (1)   and  (2)   , the better the IFPR   from the viewpoint of the multiplicative consistency.As different IFPRs   ( = 1, 2, . . ., ) 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 [0, 1]-valued interval weight vector directly from individual IFPRs: As  −  +  +  = 1 and  +  +  −  = 1 for all ,  = 1, 2, . . ., ,  = 1, 2, . . ., , we have Thus, solutions to model (23) are determined by solving the following geometric least square model: Solving this model, a group [0, 1]-valued interval weight vector is determined as

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  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  IFPRs will have to be determined.
In group decision analysis, if  IFPRs are the same, it is logical to assign their importance the same weights; that is,   = 1/ for all = 1, 2, . . ., .In this case, model ( 25) is reduced to (15).If the IFPR   is much different from the others, its importance weight should be small and the geometric mean of the difference ratios between   and the others is large.Conversely, if   is very similar to the others, its importance should be high and the geometric mean of the difference ratios between   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.
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.
Based on the above analyses, we now develop an approach for deriving a group [0, 1]-valued interval weight vector directly from individual IFPRs with unknown importance weights.The approach is described in the following steps.
Step 9.As per the decreasing order of   , a ranking order for all decision alternatives is derived, and "  being preferred to

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 ( 1 ), cost and customization ( 2 ), reliability ( 3 ), compatibility ( 4 ), and market position and reputation ( 5 ).
Although there are many potential ERP software products in the market, only five of them, denoted by  1 ,  2 ,  3 ,  4 , and  5 , are identified as candidates.A committee consisting of three experts ( 1 ,  2 , and  3 ) 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.
As the importance weights of the five criteria are to be determined, each expert   ( = 1, 2, 3) compares each pair of the criteria and provides his/her judgments by means of an IFPR  ] .
(30) On the other hand, the importance of the three experts is also unknown.Therefore, we need firstly to determine the relative weights of     3, where the first row gives the criterion weights  *   ( = 1, 2, . . ., 5) derived earlier.Similar to the treatment in Wang and Li [15], the following linear programs given by Bryson and Mobolurin [18] 3.

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 [0, 1]-valued interval weights, multiplicatively consistent interval judgments, and planes.A geometric least square model has been developed for deriving [0, 1]-valued interval weights from any IFPR and extended to generate a group [0, 1]-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.

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

Table 1 :
A comparative study for the IFPR .

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