In group decision-making (GDM), traditional consensus models have primarily focused on cardinal consensus. In reality, irrespective of whether the objective of GDM is to select the optimal alternative or to rank alternatives, it is imperative to establish a ranking that garners the utmost assent from all decision-makers (DMs). When preferences are articulated through fuzzy preference relations (FPRs), cardinal information emerges in numerical form, quantifying the degree of preference for alternatives, while ordinal relations are implicitly embedded within pairwise comparisons. To delve into both cardinal and ordinal consensus among DMs, this study introduces two consensus optimization models that strive to minimize adjustments to FPRs while fostering consensus in terms of preference intensity and ranking. To this end, we first propose two ordinal consensus measurement methods: one precisely discerns whether DMs have achieved consensus on the selection of the best alternative, while the other assesses the consistency of different preference rankings, taking into account the importance of positions. Based on these methods, two systems of inequalities are designed to explicitly govern both types of ordinal consensus. Subsequently, two consensus control rules are formulated, tailored to distinct objectives. These rules necessitate not only cardinal consensus among all DMs, but also their alignment in terms of either the selection of the best alternative or the preference ranking. Ultimately, these rules are integrated as constraints into two mixed-integer programming models aimed at minimizing preference adjustments. The proposed models have been applied in a case study, confirming their practicality, with thorough comparative analyses demonstrating their effectiveness.

Handing computing assets to cloud service providers (CSPs) to obtain cloud services is one of the important strategies for enterprises to embrace the digital era, and CSP selection is a crucial decision-making process for cloud deployment. However, there are many criteria involved in selecting an optimal CSP, not all of which can be accurately quantified. Therefore, CSP selection is a typical hybrid-information decision-making problem, in which criterion evaluation values are expressed in various forms. Meanwhile, the psychological behavior of the CSP selection team also has a significant impact on the decision-making result, which is poorly considered in the existing research results on CSP selection. Thus, in this paper, a new group decision-making support framework incorporating regret theory is constructed to select CSPs with hybrid information. Initially, various forms of hybrid information are processed separately to avoid the distortion of heterogeneous information caused by traditional conversion methods. Then, considering the psychology of regret aversion, the respective regret–rejoice functions for hybrid information are defined. Subsequently, regret–rejoice values are introduced into the evaluation of mixed data method framework, and a decision-making support procedure based on it is established, in which an expert weight determination method based on the maximizing consensus model is proposed, and the group best–worst method is used to calculate criteria weights. Afterwards, an illustrative example of CSP selection is given to clarify the implementation process of the proposed method. Finally, the effectiveness and superiority of the proposed decision-making framework in selecting CSPs are explained through parameter analysis and comparison with existing methods.

In practical applications, particularly in flexible manufacturing systems, there is a high level of uncertainty. A type-2 fuzzy logic system (T2FS) has several parameters and an enhanced ability to handle high levels of uncertainty. This article proposes an improved artificial immune system (IAIS) algorithm to solve a special case of the flexible job shop scheduling problem (FJSP), where the processing time of each job is a nonsymmetric triangular interval T2FS (IT2FS) value. First, a novel affinity calculation method considering the IT2FS values is developed. Then, four problem-specific initialization heuristics are designed to enhance both quality and diversity. To enhance the exploitation abilities, six local search approaches are conducted for the routing and scheduling vectors, respectively. Next, a simulated annealing method is embedded to accept antibodies with low affinity, which can enhance the exploration abilities of the algorithm. Moreover, a novel population diversity heuristic is presented to eliminate antibodies with high crowding values. Five efficient algorithms are selected for a detailed comparison, and the simulation results demonstrate that the proposed IAIS algorithm is effective for IT2FS FJSPs.

A q-rung orthopair uncertain linguistic set can be served as an extension of an uncertain linguistic set (ULS) and a q-rung orthopair fuzzy set, which can also be treated as a generalized form of the existing intuitionistic ULS and Pythagorean ULS. The new linguistic set uses the uncertain linguistic variable to express the qualitative evaluation information and allows decision makers to provide their true views freely in a larger membership grade space. In this paper, we investigate the Bonferroni mean under the q-rung orthopair uncertain linguistic environment, then we propose the q-rung orthopair uncertain linguistic Bonferroni mean and its weighted form. Furthermore, considering the specific partition pattern among the attributes, the q-rung orthopair uncertain linguistic partitioned Bonferroni mean and its weighted form are developed. Meanwhile, we discuss several representative cases and attractive properties of our proposed operators in depth. Subsequently, a novel multi-attribute decision-making method is developed based on the above-mentioned aggregation operators. In the end, a comprehensible case is performed to analyze the superiority of the developed method by comparing with other typical studies.

q-Rung orthopair fuzzy sets (q-ROFSs), originally presented by Yager, are a powerful fuzzy information representation model, which generalize the classical intuitionistic fuzzy sets and Pythagorean fuzzy sets and provide more freedom and choice for decision makers (DMs) by allowing the sum of the q t h power of the membership and the q t h power of the nonmembership to be less than or equal to 1. In this paper, a new class of fuzzy sets called q-rung orthopair uncertain linguistic sets (q-ROULSs) based on the q-ROFSs and uncertain linguistic variables (ULVs) is proposed, and this can describe the qualitative assessment of DMs and provide them more freedom in reflecting their belief about allowable membership grades. On the basis of the proposed operational rules and comparison method of q-ROULSs, several q-rung orthopair uncertain linguistic aggregation operators are developed, including the q-rung orthopair uncertain linguistic weighted arithmetic average operator, the q-rung orthopair uncertain linguistic ordered weighted average operator, the q-rung orthopair uncertain linguistic hybrid weighted average operator, the q-rung orthopair uncertain linguistic weighted geometric average operator, the q-rung orthopair uncertain linguistic ordered weighted geometric operator, and the q-rung orthopair uncertain linguistic hybrid weighted geometric operator. Then, some desirable properties and special cases of these new operators are also investigated and studied, in particular, some existing intuitionistic fuzzy aggregation operators and Pythagorean fuzzy aggregation operators are proved to be special cases of these new operators. Furthermore, based on these proposed operators, we develop an approach to solve the multiple attribute group decision making problems, in which the evaluation information is expressed as q-rung orthopair ULVs. Finally, we provide several examples to illustrate the specific decision-making steps and explain the validity and feasibility of two methods by comparing with other methods.

A Q-rung orthopair fuzzy set (q-ROFS) originally proposed by Yager (2017) is a new generalization of orthopair fuzzy sets, which has a larger representation space of acceptable membership grades and gives decision makers more flexibility to express their real preferences. In this paper, for multiple attribute decision-making problems with q-rung orthopair fuzzy information, we propose a new method for dealing with heterogeneous relationship among attributes and unknown attribute weight information. First, we present two novel q-rung orthopair fuzzy extended Bonferroni mean (q-ROFEBM) operator and its weighted form (q-ROFEWEBM). A comparative example is provided to illustrate the advantages of the new operators, that is, they can effectively model the heterogeneous relationship among attributes. We prove that some existing known intuitionistic fuzzy aggregation operators and Pythagorean fuzzy aggregation operators are special cases of the proposed q-ROFEBM and q-ROFEWEBM operators. Meanwhile, several desirable properties are also investigated. Then, a new knowledge-based entropy measure for q-ROFSs is also proposed to obtain the attribute weights. Based on the proposed q-ROFWEBM and the new entropy measure, a new method is developed to solve multiple attribute decision making problems with q-ROFSs. Finally, an illustrative example is given to demonstrate the application process of the proposed method, and a comparison analysis with other existing representative methods is also conducted to show its validity and superiority.

The q-rung orthopair set (q-ROFSs) can serve as a generalization of the existing orthopair fuzzy sets, including intuitionistic fuzzy sets and Pythagorean fuzzy sets. The most desirable characteristic of q-ROFSs is that they support a greater space of allowable membership grades and provide decision makers more freedom in describing their true opinions. As a classical aggregation operator, Heronian mean (HM) can model the interrelationship between attributes. In this paper, we extend the traditional HM to aggregate q-rung orthopair fuzzy information and propose the q-rung orthopair fuzzy HM and its weighted form. Further, to overcome the shortcomings of the traditional HM, considering the possible partition structure in the actual decision situations, we propose the q-rung orthopair fuzzy partitioned Heronian mean operator and the q-rung orthopair fuzzy weighted partitioned Heronian mean operator. Then, some special cases and some desirable properties are investigated and discussed. A new multiple attribute group decision-making(MAGDM) technique is developed based on the proposed q-rung orthopair fuzzy operators. Finally, a representative example is provided to verify the effectiveness and superiority of the proposed method by comparing with other several existing representative MAGDM methods.

The power average (PA) operator and Maclaurin symmetric mean (MSM) operator are two important tools to handle the multiple attribute group decision-making (MAGDM) problems, and the combination of two operators can eliminate the influence of unreasonable information from biased decision makers (DMs) and can capture the interrelationship among any number of arguments. The Pythagorean fuzzy linguistic set (PFLS) is parallel to the intuitionistic linguistic set (ILS), which is more powerful to convey the uncertainty and ambiguity of the DMs than ILS. In this paper, we propose some power MSM aggregation operators for Pythagorean fuzzy linguistic information, such as Pythagorean fuzzy linguistic power MSM operator and Pythagorean fuzzy linguistic power weighted MSM (PFLPWMSM) operator. At the same time, we further discuss the properties and special cases of these operators. Then, we propose a new method to solve the MAGDM problems with Pythagorean fuzzy linguistic information based on the PFLPWMSM operator. Finally, some illustrative examples are utilized to show the effectiveness of the proposed method.

An efficient 3D visualization system has not only fast volume rendering algorithms but also effective navigation methods. Rendering speed is one of key technologies in most 3D visualization applications. We exploit software, Pentium 4 and graphics hardware technologies, such as threshold segmentation, Intel SIMD and 3D texturing, to obtain interactive volume rendering on a standard PC without specialized expensive hardware. Path planning is essential in many 3D visualization applications, such as virtual endocopy, in order to accelerate exploring. There are three major types of methods to extract the navigation path from a 3D data set, including manual, 3D distance transform and thinning based techniques. 3D thinning is a desirable method to extract skeletons of objects, but it has some severe problems to be solved. It is time consuming with discontinuity and small branches. An effective encoding and coordinates transform based scheme is presented to generate a look up table of 3D thinning templates to speed up path extracting, and a two-pass tracking technique is followed to trim the small branches of skeletons. Tri-pass cubic Bezier technique is proposed to decrease large curvatures caused by discrete representation of path. A smooth and C 1 continuity navigation path is thus produced by our algorithms. Following this path, the camera moves and rotates smoothly without any dithering. Our system is very useful and can be widely applied due to full utilization of the existing inexpensive capabilities of PCs.