This study proposes some standard and general forms of induced ordered weighted averaging (GnIOWA) operators where the inductive information is ordered weighted averaging (OWA) weight vectors instead of real numbers. It shows the usefulness of such type of generalized induced OWA in decision-making and evaluation and many other applications. We propose three weights allocation methods that are specifically designed for the proposed GnIOWA operators. For each of the proposed weights allocation methods, a numerical example is also attached accordingly. With the use of convex/concave Regular Increasing Monotone quantifiers, we further discuss some mathematical properties of these weights allocation methods.

This study first revamps Yager prioritized ordered weighted averaging operators, and condenses them into a conceptual frame with putting aside one realization from Yager's original proposal. Then, based on elicited conceptual frame called Yager prioritized preference conceptual frame, this article proposes three distinct realizations to the conceptual frame with corresponding different instances, in which some evaluation models with weights determination methods are provided. Numerical examples are also presented immediately after every instance. Lastly, this study proposes the concepts of outer monotonic, inner monotonic, and global monotonic weights functions, and discusses some related properties, which are often embodied in preferences of decision makers.

Hesitance is an innate human psychological phenomenon pervasive in our daily life. Although it is very complex and still not fully understood, there have been many studies related to this interesting topic. In this study, we proposed two fundamental problems about hesitance: the first is the generating problem, which is related to how to elicit hesitance information from human thinking; the second is the measuring problem, which is related to how to devise effective and reasonable methods to measure hesitance degrees from given hesitance information. With these two fundamental questions, we first discussed and analyzed several examples in real life involving hesitance, showing that it can indeed be recorded and represented. Then we roughly classified the hesitance information into two major classes: static hesitance and dynamic hesitance. Some simple and interesting methods were proposed to elicit hesitance, and more reasonable methods were proposed to measure the hesitance degree from the two classes of hesitance information. All methods that have been discussed in this study try to avoid complexity in every aspect while reserving strictness and reasonability; therefore, the present study provides suitable information for practitioners with different backgrounds.

Existing extensions to Yager's ordered weighted averaging (OWA) operators enlarge the application range and to encompass more principles and properties related to OWA aggregation. However, these extensions do not provide a strict and convenient way to model evaluation scenarios with complex or grouped preferences. Based on earlier studies and recent evolutionary changes in OWA operators, we propose formulation paradigms for induced OWA aggregation and a related weight function with self-contained properties that make it possible to model such complex preference-involved evaluation problems in a systematic way. The new formulations have some recursive forms that provide more ways to apply OWA aggregation and deserve further study from a mathematical perspective. In addition, the new proposal generalizes almost all of the well-known extensions to the original OWA operators. We provide an example showing the representative use of such paradigms in decision-making and evaluation problems.

Aggregation functions are mostly used in decision-making situations that require information fusion in a meaningful manner. The main purpose of aggregation is to turn a group of input data into a single and comprehensive one. However, in real decision-making and system evaluation problems, the decision maker may exhibit only some amount of certainty in her decision inputs. In this study, we show how to aggregate these certainty degrees assigned to a group of inputs in an intuitive and reasonable manner. One of the interesting aspects of the problem is that the value aggregation is independent of their certainty degrees while the certainty aggregation essentially depends on both the input values and the value aggregation function. The construction of the aggregation function gives rise to a fuzzy measure that satisfies some very interesting properties. The technique presented here has wide range of applications.

Regular Increasing Monotone (RIM) quantifier and Ordered Weighted Function are important counterparts of discrete ordered weighted averaging operators. Some important characteristics such as entropy, Moment, and Step/Hurwicz degree have already been proposed and studied by several researchers. The main propose of this paper is to put the concepts of entropy, Moment, and Step/Hurwicz degree for RIM quantifier into a continuous environment. Some well-defined representative families of RIM quantifiers are also presented. The metric spaces of RIM quantifiers are discussed.

We study the properties of OWA multiplication monoid. By introducing and-accumulation vectors of OWA operators, we consider the set of all n-dimensional OWA operators as a lattice. Then, we analyze some monotonicity properties of OWA operators based on an ordering induced by and-accumulation vectors. We also show that the lattice-theoretical operations are a kind of counterpart of the OWA multiplication monoid (and its dual, additive monoid). An example of using the OWA multiplication monoid and the lattice-theoretical structure in decision making problem is provided.

Ordered weighted average (OWA) operators with their weighting vectors are very important in many applications. We show that directly taking Minkowski distances (including Manhattan distance and Euclidean distance) as the distances for any two OWA operator is not reasonable. In this study, we propose the standard distance measures for any two OWA operators and then propose a standard metric space for the set of all n-dimension OWA operators. We analyze and discuss some properties of the introduced OWA metric and further propose a metric space of Choquet integrals represented by the underlying fuzzy measures. Some applications in decision making of OWA distances are also presented in this study.

Step-like or Hurwicz-like ordered weighted averaging (OWA) (S-H OWA) operators connect two fundamental OWA operators, step OWA operators and Hurwicz OWA operators. S-H OWA operators also generalize them and some other well-know OWA operators such as median and centered OWA operators. Generally, there are two types of determination methods for S-H OWA operators: One is from the motivation of some existed mathematical results; the other is by a set of “nonstrict” definitions and often via some intermediate elements. For the second type, in this study we define two sets of strict definitions for Hurwitz/step degree, which are more effective and necessary for theoretical studies and practical usages. Both sets of definitions are useful in different situations. In addition, they are based on the same concept moment of OWA operators proposed in this study, and therefore they become identical in limit forms. However, the Hurwicz/step degree (HD/SD) puts more concerns on its numerical measure and physical meaning, whereas the relative Hurwicz/step degree (rHD/rSD), still being accurate numerically, sometimes is more reasonable intuitively and has larger potential in further studies and practical applications.

In this study, we propose the concept of piled ordered weighted averaging (OWA) operators, which generalize the centered OWA operators and also connect the step OWA operators with the Hurwicz OWA operators with given the orness degree. We propose a controllable algorithm to generate the family of piled OWA operators depending on their predefined three parameters: orness degree, step-like or Hurwicz-like degree, and the numbers of “supporting” vectors. By these preferences, we can generate infinite more piled OWA operators with miscellaneous forms, and each of them is similar to the well-known binomial OWA operator, which is very useful but only has one form corresponding to one given orness degree.