{'fraction_non_alphanumeric': 0.02391954335417233, 'fraction_numerical': 0.002989942919271541, 'mean_word_length': 5.536412078152753, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Determination of Criteria Weights Based on Soft Sets and Aggregation Operators under Preference Orderings in MCGDM

Preference information integration of experts in weights determination of criteria is principle in MCGDM (Multiple Criteria Group Decision Making). Based on the nonrestraint of parameters selection of soft sets and the characteristic of specified aggregation operators, this paper describes a novel approach of weight determination under preference orderings of different experts. The corresponding algorithm is given and implemented by Matlab. Lastly, an illustrative example is given to determine the weights of criteria in the partner selection of R&D cooperation of enterprises. The result shows that the approach is efficient and feasible. Introduction Many Group decision-making issues are often determined by several relative decision-makers and experts. In multiple criteria group decision making (MCGDM), decision makers (DMs) always give preferences information on alternatives or decision criteria. Because of the difference of cultural and educational background, familiarity degree with alternatives, evaluating level, preferences of experts may be different. For reflecting the difference of different experts or decision criteria, weights of criteria and experts are critical in MCGDM. The approaches for weights determination of criteria come into three categories: subjective methods of weights determination, such as Delphi method; objective methods of weights determination, such as principal component analysis; the synthetic methods which means the combination of subjective and objective methods, such as analytic hierarchy process-AHP. In the process of determining weights of criteria or experts, there exists nonstandard, stochastic, fuzzy and subjective data. Z. S. Xu focused on multiple aggregation operators to aggregate special preference information, such as linguistic aggregation operators applied for linguistic preference relations[1], the hybrid operator generalizing for reflecting the importance of both the given argument and the ordered position of the argument[2]. Some new models are provided to deriving the weights of experts or criteria based on different preference presentations. A distance-based aggregation approach was proposed to assess the relative importance weights for GDM with interval preference orderings[3]. A belief structure was also provided to represent the partially ordered preferences with belief degrees, which can cover both qualitative and quantitative aspects of the evaluation[4]. Das S. et al. proposed an objective method for determining weights of criteria based on a new measure of intuitionistic fuzzy information[5]. Furthermore, a new method determining the unknown preferences for group decision making was presented to satisfy order consistency for information aggregation[6]. A soft set[7] is a parameter set over a subset of an initial universe set. Researchers can choose the form of parameters they needed, which greatly simplifies the decision-making process and make the process more efficient. Based on the work of Molodtsov, many scholars have done further research on soft sets theory. P. K. Maji et al. defined equality of two soft sets, subset and super set of soft set and so on[8]. The applications of soft sets theory are also extended to different domains, such as stock market analysis[9] and suppliers selection[10]. In processes for making decision, several experts may be needed to determine weights of criteria. If each expert gives an order relation of all evaluation criteria without giving detailed scores or grades of criteria, the soft sets will be a perfect