Weight assignment method for multiple attribute decision making with dissimilarity and conflict of belief distributions
Graphical abstract
Introduction
Multiple attribute decision making (MADM) problems (Dong et al., 2019, Liang et al., 2018, Liu et al., 2019) are usually comprised of a set of attributes, which can be either quantitative or qualitative. Qualitative attributes can be expressed by various forms, such as linguistic variable (Gou et al., 2018, Wu et al., 2019), belief distribution (BD) (Dubois and Prade, 1988, Fu et al., 2018), intuitionistic fuzzy set (Chen, 2014, Li and Deng, 2019, Milošević et al., 2017); hesitant fuzzy sets (Xue, Xu, Wang, & Ren, 2019); and so on. The evidential reasoning (ER) approach (Akhoundi and Nazif, 2018, Wang et al., 2006, Xiao, 2019, Xu et al., 2016, Xu et al., 2017, Yang and Xu, 2002, Yang and Xu, 2013, Yang, 2001, Zhou et al., 2018, Zhou et al., 2019) provides a probabilistic aggregation process for MADM problems where assessments are represented by BDs, and uncertainty, incompleteness and fuzziness can all be dealt with in a consistent way. Since attribute weight is an important factor in the aggregation process of a MADM problem, how to generate a set of rational and valid weights is significant for the ER approach or other evidence combination methods (Xiao, 2019, Yin et al., 2019). Weight assignment methods can be broadly classified into three categories: subjective, objective and hybrid (Fu and Wang, 2015, Wang and Luo, 2010). Subjective method includes direct rating method (Bottomley & Doyle, 2013), weighted least square method (Chu, Kalaba, & Spingarn, 1979), Delphi (Hwang & Lin, 1987) and so on. This kind of method extracts the attribute weights directly from the decision maker (DM) through interview, discussion or questionnaire. Objective method generates attribute weights from the intrinsic information of the assessment values. It can be classified into two sub-categories. One is based on the divergence of values from the assessment of different alternatives on each attribute. Representative methods include the entropy weight assignment method (EWAM) (Song et al., 2017, Zhou et al., 2019), standard deviation (SD) (Chin et al., 2015, Diakoulaki et al., 1995), maximizing deviation method (Qian & Luan, 2017) and discriminating power method (Fu et al., 2018). The other sub-category not only depends on the dimension of information extracted in the first category, but also the correlation between each pair of attributes that reflects interdependency or conflict of attributes. Typical methods include criteria importance through intercriteria correlation (CRITIC) (Diakoulaki et al., 1995), correlation coefficient and standard deviation integrated (CCSD) method (Wang & Luo, 2010), deviation and decision incompatibility based method (Chin et al., 2015), and so on. Hybrid method (Yang, Yang, Xu, & Khoveyni, 2017) is applied when both subjective judgment and numerical evaluation values can be obtained.
Although a great number of methods for assigning attribute weights have been proposed in the last few decades, how to elicit appropriate weights from subjective assessments on attributes still remains an open issue, especially if there is no prior knowledge. The above mentioned objective weight assignment methods such as EWAM, SD method, CRITIC, CCSD and maximizing deviation method all assume that each attribute is assessed by a numerical value no matter it is in a quantitative or qualitative nature. In real decision-making problems, qualitative attributes are usually expressed by subjective judgments, e.g., BD, hesitant fuzzy sets. If there is no prior knowledge on the importance of attributes, the method of deriving attribute weights from the subjective assessments together with numerical values on quantitative attributes needs to be studied comprehensively. In recent years, some research has been devoted on generating attribute weights provided that qualitative attributes are denoted by BDs, such as the above mentioned discriminating power method. But they only considered one dimension of information which reveals the discrepancy among the BDs of different alternatives associated with a specific attribute. The other dimension which measures the interdependency between the BDs of each pair of attributes was not considered in the previous studies. In either the CRITIC or CCSD, the interdependency is quantified by the Spearman correlation coefficient among attributes which are all represented by numerical values. When the values of attributes are represented by BDs, the conflict or dissimilarity measure between BDs is the basis to measure the interdependency between attributes. Up to now, many metrics have been proposed on the conflict or dissimilarity measure such as Tessem’s distance (Tessem, 1993), combined dissimilarity measure (Liu, Dezert, Pan, & Mercier, 2011); Jousselme’s distance (Jousselme, Grenier, & Bossé, 2001); cosine similarity (Wen, Wang, & Xu, 2008), correlation coefficient (Jiang, 2018), Liu’s distance (Liu, 2006). But none of them is perfect to tackling with all circumstances which will be detailed in Section 3.3. If BDs are transformed to numerical values such as utilities, some information contained in BDs may not be preserved. So how to measure the interdependency between two attributes represented by BDs is still an unanswered question. Inspired by the concept of dissimilarity measure between the BDs of DMs (Fu, Yang, & Yang, 2015) and alternatives (Fu et al., 2018) defined by Fu et al., this paper proposes the dissimilarity measure between the BDs of two attributes on an alternative. Then the conflict measure between two attributes is defined on both the alternative and evaluation grade level in order to quantify the interdependency. As a result, a novel weight assignment method which considers both the above two dimensions derived from BDs is developed in this paper. Optimization models are also constructed for the consideration of incompleteness included in BDs.
When the subjective judgments from multiple sources are aggregated to an overall assessment, how to measure the support degree of assessment for a decision is a critical issue because it may affect the likelihood of making a right decision on two aspects. One is the external divergence which can be quantified by the dissimilarity of different attributes. The other one is the internal indeterminacy which is correlated with the uncertainty measure of original subjective judgments. The uncertainty measure of belief distributions has widely accepted solutions (Deng, 2016). In the past decades, a lot of uncertainty measures have been proposed, e.g., Klir & Ramer’s discord (Klir & Ramer, 1990); Deng entropy (Deng, 2016); Radim & Prakash’s total uncertainty (Jiroušek & Shenoy, 2018); Jousselme’s ambiguity measure (Jousselme, Liu, Grenier, & Bosse, 2006), Yager’s interval-valued entropy (Ronald & Yager, 2018); Yang’s total uncertainty measure (Yang & Han, 2016). But they are all defined in the context of Dempster-Shafer’s evidence theory. In a decision-making problem, the utility of each focal element should be considered in the definition of the uncertain degree in a BD. For example, A is assessed to be ‘Excellent’ and ‘Average’ with the belief degree of 0.5 and 0.5, while B is assumed to be ‘Good’ and ‘Average’ with the belief degree of 0.5 and 0.5. According to any one of the above mentioned uncertainty measures on mass function, the result is identical for A and B although it may be different by each method. It is not reasonable because the difference between the utility of ‘Excellent’ and ‘Average’ is larger than ‘Good’ and ‘Average’, which leads to the divergence of opinion for A is more than B in either a GDM situation or individual decision circumstance. So the uncertainty measure on mass function should be improved to considering the difference among the utilities of discrete focal elements. In this paper, the concepts of dissimilarity measure and uncertainty measure on alternatives are proposed to generate the support degree of assessment for decision making. The uncertainty measure on the BDs of all attributes and the aggregated BD are presented, followed by the definition of incompatibility measure among the BDs of attributes for the purpose of quantifying the discrepancy of the BDs on different attributes.
The main contributions of the paper are summarized as follows:
- (1)
The conflict measures between two attributes on both the alternative and evaluation grade level are defined provided that subjective judgements are represented in the form of BDs. The advantages of the measurement are analyzed compared with existing conflict or distance metrics.
- (2)
A novel weight assignment method is developed from two dimensions extracted from the subjective judgments of attributes. This enables us to determine the weights of qualitative attributes if we are confronted with lack of prior knowledge. Comparative analysis is conducted to show the effectiveness and applicability of the proposed method.
- (3)
In order to facilitate the DM to measure the support degree of assessment for decision making in a MADM problem, the dissimilarity measure on alternatives and the uncertainty measure of BD are proposed. The concepts of average and global uncertainty measure are defined, which induce the total incompatibility measure among the BDs of all attributes for an alternative. This enables us to utilize the aggregated result in a more rational way instead of just depending on the ranking order of different alternatives.
The reminder of the paper is organized as follows. Section 2 is a brief presentation of the ER approach. In Section 3, conflict measure on the alternative level and evaluation grade level are firstly given, followed by the description of a comprehensive weight assignment method based on BDs. 4 Dissimilarity measure on alternative, 5 The uncertainty measure of BDs are both used to discuss the support degree of assessment for decision making. More specifically, Section 4 provides a method to quantify the external dissimilarity measure on alternatives, while Section 5 proposes a measurement of internal indeterminacy or uncertainty on belief structures. Section 6 presents a case study and the comparative analysis with existing methodological methods are conducted. This paper is concluded in Section 7.
Section snippets
Preliminaries
The ER approach uses the belief structure to represent uncertain subjective judgment on qualitative attribute based on the framework of Dempster-Shafer (D-S) evidence theory (Jiroušek and Shenoy, 2018, Klir and Ramer, 1990, Ronald and Yager, 2018) and decision-making theory. A set of evaluation grades for the assessment of an attribute on an alternative constitute the frame of discernment which is profiled as follows:where each denotes an evaluation grade, and
Assignment of attribute weight based on BDs
The dissimilarity measure of assessments is a critical issue to derive attribute weights in an objective way for MADM problems. When the assessments are presented in the form of numerical values, geometric models such as Euclidean distance, Manhattan distance, Chebyshev distance can be applied. Besides, Hamming distance, correlation coefficient, information entropy can also be employed to quantify the dissimilarity between objects. If the numerical values are not sufficiently informative, fuzzy
Dissimilarity measure on alternative
In a decision-making problem, the support degree of assessment for the final decision is pivotal because it determines the extent that we can rely on the aggregated result. However, if the assessment value of each alternative differs significantly over different attributes, it may be difficult to make a ranking order firmly. On the contrary, it is easier to make a choice provided that the BDs of attributes achieve a high degree of consistency no matter the assessment is positive or negative.
The uncertainty measure of BDs
The second factor to determine the support degree of assessment is the internal indeterminacy of a BD. It can be measured by the uncertainty of the original assessment provided that subjective judgments are extracted from an individual or a group of DMs. If the BDs assessed on most attributes are associated with a high degree of uncertainty, the support degree of the aggregated result will be low no matter which combination rule is used. An extreme case is that all attributes are assessed to be
Case study
It was discussed in the introduction section that a variety of methodological methods to determine attribute weights have been designed in order to handle different situations. Each of them has its unique feature which differentiates from other techniques. In this subsection, a car selection problem adapted from (Yang, 2001) is chosen to illustrate the effectiveness and applicability of the proposed method. Several different weight assignment methods are compared with the proposed method to
Conclusions
In this paper, a novel weight assignment method for MADM problems is proposed on condition that the assessments are presented by BDs. The main principle behind the method is that the weight is correlated with the divergence of subjective judgment among different alternatives and the conflict between each pair of attributes. To do that, the conflict measures on both the alternative and evaluation grade levels between two attributes are designed. A comprehensive weight assignment method is then
CRediT authorship contribution statement
Mi Zhou: Conceptualization, Methodology, Investigation, Writing - original draft. Yu-Wang Chen: Review, Grammar checking. Xin-Bao Liu: Formal analysis, Supervision. Ba-Yi Cheng: Review. Jian-Bo Yang: Supervision.
Acknowledgements
This research is supported by the National Natural Science Foundation of China under the Grant No. 71601060, 71571166 and 71971135, NSFC-Zhejiang Joint Fund for the Integration of Industrialization and Informatization under the Grant No. U1709215, Innovative Research Groups of the National Natural Science Foundation of China under the Grant No. 71521001, Natural Science Foundation of Anhui province under the Grant No. 1908085MG223.
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