Vague Number Of Incomplete Multi-criteria Decision Making Approach

There are a great many of Fuzzy Multi-criteria Decision-making (FMCDM) problems in the social life and economic activities. Vague multi-criteria decision-making problems are one kind of them which are very important. As the decision-making problem itself is fuzzy and complex, and decision-maker has limited knowledge construction, there are many fuzzy multi-criteria decision-making problems with incomplete information of the weights of decision-makers, incomplete or completely unknown the information of criterion weights and the criterion values are uncertain in the real decision-making. So it has much theoretical and practical significance to do some researches on the theory and methods of the vague multi-criteria decision-making problems with incomplete certain information.Based on the summary and analysis of research results of predecessors, and according to the equivalence of the definition of vague sets (VSs) and the definition of intuitionistic fuzzy sets (IFSs), trapezoidal vague numbers are defined and a preliminary study of the related theory is proposed. Based on these research results, Fuzzy multi-criteria decision-making methods based on traditional fuzzy numbers are extended to fuzzy multi-criteria decision-making methods based on vague numbers, and the corresponding decision-making models are established and then worked out on the basis of the latest optimization theory and optimizing algorithms. And the details are as follows:(1) Trapezoidal vague numbers and their operation are defined, and then distance between two trapezoidal vague numbers, utility and similarity are studied according to the distance, utility and similarity of vague values. In addition, the method of comparison of two vague numbers and standardized method are also studied. On this basis, fuzzy multi-criteria decision-making models in which the criterion weights are real numbers or incomplete, and the criterion values are vague numbers are established. To the fuzzy multi-criteria decision-making problem with he criterion weights being real numbers, fuzzy multi-criteria decision-making model is established based on the utility of trapezoidal vague numbers. In this method, the criterion values which are all trapezoidal vague numbers are transformed into the corresponding utility values at first, and then these utility values are aggregated respectively. Finally, the decision maker can rank the alternatives or opt for the best alternative by comparing the aggregated utility values of alternatives. To the fuzzy multi-criteria decision-making problem with the criterion weights being incomplete, fuzzy multi-criteria decision-making model based on vague similarity and model based on vague distance are established respectively. In these two models, used the vague similarity and the vague distance, the linear programming models are established respectively, and then the optimal weights of the first n criteria are gotten by solving these two models. Finally, the decision maker can rank the alternatives or opt for the best alternative by comparing the final close-degrees of alternatives or relative values of goodness.(2) Vague weighted arithmetic averaging (V-WAA) operator, ordered weighted averaging (V-OWA) operator, extended compromise V-OWA(CV-OWA) operator , extended weighted V-OWA (WV-OWA) operator, extended ordered weighted V-OWA (OWV-OWA) operator and extended vague ascending ordered weighted Euclid averaging (V-AOWEA) operator based on vague numbers are defined, and their properties are discussed. On this basis, some fuzzy multi-criteria decision making models based on these operators in which the criterion weights are real numbers or completely unknown and the criterion values are trapezoidal vague numbers are proposed. To the fuzzy multi-criteria decision-making problem with the criterion weights being certain, fuzzy multi-criteria decision-making model based on V-WAA operator and model based on extended WV-OWA operator are established respectively. To the fuzzy multi-criteria decision-making problem with the criterion weights being completely unknown, some methods (fuzzy linguistic quantitative methods, for example) are used to get the weights of the first n criteria at first, fuzzy multi-criteria decision-making models based on V-OWA operator, extended OWV-OWA operator and extended V-AOWEA operator are established respectively. In decision-making process of these models, the first n criterion values are aggregated by the corresponding operator at first, and then the final criterion value of every alternative is gotten by comparing the aggregated value of the first n criterion values and the (n+1)th criterion value. Finally, the decision maker can rank the alternatives or opt for the best alternative by comparing the final weighted utility values of alternatives or the final criterion values of alternatives.(3) Decision maker's vague confidence degree, combined weighted arithmetic averaging (V-CWAA) operator and extended weighted arithmetic averaging (V-COWA) operator are defined, and then fuzzy multi-criteria decision-making models in which the decision maker's weights and the criterion weights are all real numbers or incomplete or the decision maker's weights are real numbers and the criterion weights are completely unknown, and the criterion values are vague numbers are established. To the fuzzy multi-criteria decision-making problem with the decision maker's weights and the criterion weights being all real numbers, fuzzy multi-criteria group decision-making model based on fuzzy consistent relationship and group decision-making model based on V-CWAA operator are established respectively. To the fuzzy multi-criteria decision-making problem with the decision maker's weights and the criterion weights being all incomplete, fuzzy multi-criteria group programming model based on decision maker's risk attitude and vague confidence degrees is established. To the fuzzy multi-criteria decision-making problem with the decision maker's weights being real numbers and the criterion weights being completely unknown, fuzzy multi-criteria group decision-making model based on C-COWA operator is established.