چگونگی مسئولیت رسیدگی به عدم قطعیت در تحلیل سلسله مراتبی : تجزیه و تحلیل ابری سلسله مراتبی دلفی

In practice, many practical problems occur in uncertain environments, especially in situations that involve human subjective evaluation such as that in the analytic hierarchy process (AHP). This paper presents a practical multi-criteria group decision-making method for decision making under uncertainty. To handle the randomness and fuzziness of individual judgments, the normal Cloud model, group decision-making technique, and the Delphi feedback method are adopted. In the proposed Cloud Delphi hierarchical analysis (CDHA), experts are asked to express their judgments using interval numbers. Individual fuzziness and randomness are then mined from the interval-value comparison matrices. Subsequently, the interval-value pairwise comparison matrices are converted into the corresponding Cloud matrices, and the one-iteration Delphi process is executed to diminish individual judgment mistakes. The individual Cloud weight vectors are calculated using the geometric mean technique and are finally weighted to form the group Cloud weight vector. A simple case study that involved reproducing the relative area sizes of six provinces in China shows that the CDHA method can effectively reduce mistakes and improve decision makers’ judgments in situations that require subjective expertise and judgmental inputs. In addition, a practical decision-making problem in which houses are ranked by home buyers shows that the proposed method is effective when applied to complex, large, multidisciplinary problems with considerable uncertainties.

Analytic hierarchy process (AHP) [28], [29] and [30] is one of the most commonly used utility-based methods for multi-criteria (-attribute) decision making [41]. The AHP uses objective mathematics to process the subjective and personal preferences of an individual or a group in decision making. In Saaty’s hierarchical analysis, a person (expert, judge, etc.) is asked to provide his/her ratios aij for each pairwise comparison between issues (alternatives, candidates, etc.) Al, A2, … , An for each criterion (objective) in a hierarchy and also between the criteria. To make comparisons, a scale of numbers should be used to indicate how many times more important or dominant one element is over another element with respect to the criterion or property being compared. In Saaty’s fundamental nine-scale measurement used in making a comparison, the numbers for the ratios are usually taken from the set {1, 2, … , 9}. It consists of verbal judgments that range from equal to extreme (equal, moderately more, strongly more, very strongly more, extremely more). The numerical judgments (1, 3, 5, 7, 9) correspond to the verbal judgments and compromises between these values. For example, if a person considers A1 to be moderately more important than A2, then al2 is equal to 3/l. The ratio aij indicates the strength with which Ai dominates Aj. The nine-scale measurement is widely applied in AHP. However, the language description disagrees with the numerical values of the scale division in various aspects [47]. The first disagreement is that the numerical values are not exactly equivalent to the verbal judgments. If the sizes (usually importances) of al and a2 are compared, al2 = 3 represents that al is three times as large (important) as a2. However, the corresponding verbal judgment states that al is moderately larger (moderately more important) than a2. Usually, in our opinion, “three times as large” does not merely indicate moderately more importance. Furthermore, if al is moderately more important than a2, and a2 is moderately more important than a3, al might be considered more or strongly more important than a3. However, in Saaty’s nine-scale measurement, al3 = al/a3 = (al/a2) × (a2/a3) = 3 × 3 = 9, which means that al is extremely more important than a3. This finding reflects an inconsistency between the qualitative descriptions and their corresponding quantitative numbers. In addition, the intrinsically nonlinear subjective perceptions of human beings usually lead to conflicts and inconsistencies in AHP. The second disagreement is that the same qualitative verbal judgment has different meanings for different persons (experts, judges, etc.). One person may say that “moderately more” represents 1.2 or so, but to another person, it represents 2 or so. Sometimes, even the same person (expert, judge, etc.) may assign different meanings to the same qualitative verbal judgment in different situations. For example, in the three circles in Fig. 1a, one may say that A is moderately larger than B, but if presented with Fig. 1b, the same person may think A is strongly larger than B. However, the two As and the two Bs in Fig. 1a and b are the same. One can obtain different qualitative verbal judgment results even with the same A and B in different situations.Ensuring consistency/unison in AHP and in group decision is difficult because of the drawbacks of qualitative linguistic judgments, particularly the disagreements between the language description and the numeric relation of the nine-scale division measurement. In practice, establishing uniform linguistic term sets for different people and problems is almost impossible. Thus, a unitive scale measurement like crisp numbers should be used to conduct comparisons. However, assigning pairwise comparisons usually involves uncertainties because of the inherent subjective nature of human judgments. The complexity and uncertainty involved in real-world decision-making problems and the inherent subjective nature of human judgments pose challenges for experts in developing crisp decision-making methodologies with precise numerical values. Decision-making processes also become difficult to implement. Providing fuzzy or interval-value opinions for the judgments in a pairwise comparison matrix is easier, and a number of techniques that use a fuzzy or interval comparison matrix to generate weights have been developed. The earliest work in fuzzy AHP was conducted by Van Laarhoven and Pedrycz [37]. In their work, the triangular fuzzy number of the fuzzy set theory was brought directly into the pairwise comparison matrix of the AHP. Buckley [4] used fuzzy numbers instead of exact (crisp) numbers and utilized the geometric mean method to calculate fuzzy weights. In the proposed fuzzy hierarchical analysis (FHA), ratio aij is expressed as approximately 5/1 instead of exactly 5/l, or that a ratio is between 4/1 and 6/1 instead of exactly 5/l. The membership functions for the final fuzzy weights of FHA can be shown graphically, allowing an intuitive ranking of the alternatives. However, the membership functions are nonlinear, and clear parameters that have certain physical meanings to denote the fuzziness and uncertainty of the final fuzzy weights do not exist. Chang [6] introduced a new approach for handling fuzzy AHP using triangular fuzzy numbers for pairwise comparisons and the extent analysis method for the synthetic extent values of the pairwise comparisons. Chang’s fuzzy AHP derives crisp weights for fuzzy comparison matrices. The computational simplicity of this approach has attracted a number of applications. However, Wang et al. [39] recently proved that such a method was found to be unable to derive the true weights from a fuzzy or crisp comparison matrix. Moreover, the weights determined using the extent analysis method did not represent the relative importance of decision criteria or alternatives at all. Buckley et al. [5] directly fuzzified Saaty’s original procedure of computing weights in hierarchical analysis to obtain fuzzy weights in the FHA. However, solving a series of (they used α -cuts) nonlinear optimization fuzzy models is computationally complicated because these models usually require intelligent algorithms such as evolutionary algorithm [5] and integrate simulated annealing algorithm, neural network, and fuzzy simulation techniques [22]. Csutora and Buckley [11] developed the lambda-max method, which is the direct fuzzification of the well-known λmax method. This method also entails transforming a fuzzy comparison matrix into a series of interval comparison matrices using α -level sets and the extension principle. Therefore, the method requires the solution of a series of eigenvalue problems. Wang et al. [40] proposed an eigenvector method (EM) to derive a normalized interval or fuzzy eigenvector from an interval or fuzzy comparison matrix. However, the EM is not applicable when there is a weight greater than or equal to 0.5. All the aforementioned approaches employ type-1 fuzzy sets, such as triangular fuzzy numbers or trapezoidal fuzzy numbers, to make the comparisons in AHP. However, type-1 fuzzy sets quantify the membership degree of an element as an accurate value between 0 and 1. In practice, the exact membership function does not exist and comprises some uncertainties and fuzziness. Saying that something certain can model something uncertain is contradictory [44]. Therefore, Sadiq and Tesfamariam proposed the intuitionistic fuzzy analytic hierarchy process (IF-AHP) [34] to handle both the vagueness and the ambiguity type of uncertainties in establishing pairwise comparisons. To avoid computational complexity, they used fixed values of fuzzification factors Δμ and degrees of belief for different pairwise comparisons. Their approach is inconsistent in practice because different comparisons and persons have various fuzziness and uncertainties. Recently, Dong et al. extended the AHP using the 2-tuple fuzzy linguistic model [14]. Intuitionistic fuzzy sets are equivalent to interval-valued fuzzy sets and type-2 fuzzy sets. In this paper, the judgments are expressed using the interval approach [23], and the uncertainties of the comparisons are modeled using the normal Cloud model. Saaty and Vargas [32] first proposed interval judgments for the AHP method as a way to model subjective uncertainty. They used a Monte Carlo simulation approach to determine weight intervals from interval comparison matrices. Since this proposal, different approaches have been developed to derive priorities from interval comparison matrices. Arbel [1] viewed interval judgments as linear constraints on weights and formulated the weight estimation problem as a linear programming (LP) model. Arbel and Vargas [2] then applied a uniform distribution to the intervals selected by decision makers (DMers) and established a connection between the Monte Carlo simulation and the LP model. Haines [16] proposed a statistical approach to extract preferences from interval comparison matrices. Two specific distributions on a feasible region were examined, and the mean of the distributions was used as a basis for assessment and ranking. Subsequently, many other simulation approaches have been reported for the interval AHP. Zhang et al. [49] adopted both normal and uniform distributions, while Banuelas and Antony [3] used gamma and triangular distributions. The computation for the conventional continuous simulation approaches is complicated and time consuming. Therefore, Cox [10] proposed a discrete simulation and a complete enumeration approach by employing elements from the set I = the integers between 1 and 9 and their reciprocals. However, the accuracy of the resultant priority intervals may not be satisfactory because of the limited number and scope of set I for simulations. Aside from the simulation approaches, numerous programming models for interval AHP are available. Sugihara et al. [36] proposed an interval regression analysis approach for obtaining interval weights from both crisp and interval comparison matrices. Mikhailov [26] proposed a fuzzy preference programming method for deriving crisp priorities from interval comparison matrices. Wang et al. [42] proposed a two-stage logarithmic goal programming method to generate weights from interval comparison matrices. Wang et al. [43] also designed a consistency testing method to check whether an interval comparison matrix is consistent. Recently, Guo et al. [15] elicited interval probabilities from interval comparison matrices using LP models. Pedrycz and Song [27] treated the entries of the reciprocal matrices not as single numeric values but rather as information granules modeled by intervals to attain a higher consensus level in the AHP group decision-making scenario. A composite index was defined using an additive combination of individual consistency and group consensus. The composite index was subsequently optimized using the particle swarm optimization method. Providing fuzzy or interval judgments for a part of or for all the judgments in a pairwise comparison matrix, such as in the aforementioned fuzzy/interval AHPs, may be more natural or easier. However, the mathematical complexity involved in the process may restrict their practicability. In addition, many published articles contend that a user might have different confidence levels on different (pairwise comparison) judgments. Nevertheless, these articles neither ask a user about how certain he/she is nor study whether changing the 1–9 scale value to match the user’s confidence improves the validity of the outcome [33]. Instead, numerous “fuzzy” judgments of fuzzy AHPs are at the same confidence level and can be represented by crisp numbers, such as [8], [12], [34] and [38]. For example, the linguistic scales “equally important,” “weakly important,” “essentially important,” “very strongly important,” and “absolutely important” are represented by fixed triangular fuzzy numbers (1, 1, 3), (1, 3, 5), (3, 5, 7), (5, 7, 9), and (7, 9, 9), respectively. “Note that, if all of the judgments are at the same confidence level and can be represented by crisp numbers, there is no point to using interval or fuzzy numbers. The point is to use fuzzy numbers as a suitable tool to represent uncertain judgments, rather than to ‘fuzzify’ certain and crisp judgments [33]”. We are in general agreement with Saaty and Tran on these points; however, our view is that DMers usually feel more comfortable and confident with providing interval judgments rather than with expressing their judgments as single numeric values. Different judgments should be represented at different granularities and confidence levels. The expert’s uncertainties should then be mined from his/her interval comparison judgment matrix to identify and utilize the information on uncertainties.

The CDHA process is proposed to handle both individual (intrapersonal) and group (interpersonal) judgment uncertainties in AHP. It has the following features: (1) The intrapersonal uncertainties are modeled by the Cloud model. The Cloud model has excellent properties including computational simplicity and natural similarity with the uncertainties in subjective cognition. The numerical parameters of the Cloud model make it intuitive to understand and easy to calculate. We adopt the interval approach to collect DMers’ opinions, which makes them express fuzzy and uncertain judgments more easily. Then, we automatically obtain the intrapersonal fuzziness and randomness from interval comparison judgments. The Cloud model can be exhibited in a graphical way so that the later stages, such as intercomparison, re-examination, and analysis, become more straightforward. (2) The group decision-making technique is adopted to handle the interpersonal uncertainty and to overcome the knowledge limitation of individuals. Different weights are assigned to different DMers according to the quality of their comparison matrices and to their professional knowledge background. (3) A one-iteration feedback and re-evaluation Delphi method is executed to reduce individual mistakes (intrapersonal randomness). In the study, the DMers received feedback information about the group judgment and the consistency of their pairwise comparisons immediately. They had therefore the possibility to redo them. After interacting with group members, individuals can modify their preference information gradually in the process of decision making to make the decision result more reasonable. (4) All the calculations are easy to implement without any intelligent algorithm. This method is very feasible in practical applications. The simple case study of reproducing the relative area sizes of six provinces in China shows that the CDHA process can reduce mistakes and uncertainties, and thus improve DMer’s subjective judgments. A practical decision-making problem of ranking houses by home buyers also shows the effectiveness of the proposed method in real-life decision problems with considerable uncertainties. The capability of handling uncertainties efficiently using simple computations makes CDHA very suitable for a number of applications, such as multi-criteria (-attributes) evaluation, ranking, selection, optimization, prediction, and decision making under uncertainty in a group scenario and in diverse areas. Future research will be concerned with more practical applications using CDHA, and modeling words and constructing the system of “computing with words” using Cloud models.