Complement Operation Research Articles

Study on multidimensional fuzzy graphs through modified partial ordering

This paper introduces the concepts of multidimensional fuzzy graphs and edge-powered multidimensional fuzzy graphs, which employ a hybrid structure that combines multidimensional fuzzy sets and graphs. This study redefines the axioms of multidimensional $t-$ norms and $t-$ conorms by providing a more general partial order that can link more components of the range set $\mathcal{J}_{\infty}\big([0,1]\big)$. More studies on various operations such as direct product, composition, tensor product, join, etc. are conducted with relevant illustrations. A novel complement operator approach is also investigated to link the multidimensional fuzzy graph and the edge-powered multidimensional fuzzy graph. Finally, defining the infimum and supremum of an arbitrary family in $\mathcal{J _{\infty}\big([0,1]\big)$ introduces many notions such as vertex degree, $min-$ vertex degree, $max-$ vertex degree, path strength, etc.

Quintic fuzzy sets: A new class of fuzzy sets for solving multi-criteria decision-making problems under uncertainty

This study proposes a new class of fuzzy sets called Quintic Fuzzy Set (QuFS), where the sum of the fifth power of the membership degree and non-membership degree is less than or equal to one. We introduce the complement operator and basic operations on QuFS and further develop various theorems based on these fundamental set operations. We also present the standard modal operators on QuFS and discuss its elementary properties. We then define a novel score and accuracy function for ranking QuFS, as well as the Euclidean and Hamming distances between two QuFSs. A supplier selection example is presented in detail to illustrate the viability and utility of the proposed QuF-TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) method. We also present a multi-criteria decision-making (MCDM) problem on COVID-19 lab selection to perform a comparative analysis of QuFS with Fermatean Fuzzy Set (FFS) and Pythagorean Fuzzy Set (PFS). Furthermore, we demonstrate an existing real case study on selection of Health care waste treatment (HCWT) technique in India and perform a comparative analysis with TOPSIS, SAW (Simple Additive Weighting) and WASPAS (Weighted Aggregated Sum Product Assessment) MCDM methods in QuF framework. With the advent of QuFS, a decision-maker can now employ the fuzzy notion to manage complicated and uncertain human judgements. The comparative results indicate that the suggested QuF measures and QuF MCDM technique outperform other fuzzy concepts regarding effectiveness and validity. Due to the restrictive nature of the current fuzzy set theories, the decision-makers in some complex, uncertain real-life problems are not free to choose the degree of membership and non-membership of their choice. Compared to the Intuitionistic Fuzzy Set (IFS), PFS, and FFS, the QuFS developed in this study has a bigger space grade space. This feature of QuFS enables decision-makers to structure complex, uncertain circumstances with greater flexibility in terms of membership and non-membership degrees.

Ill-Typed Programs Don’t Evaluate

We introduce two-sided type systems, which are sequent calculi for typing formulas. Two-sided type systems allow for hypothetical reasoning over the typing of compound program expressions, and the refutation of typing formulas. By incorporating a type of all values, these type systems support more refined notions of well-typing and ill-typing, guaranteeing both that well-typed programs don't go wrong and that ill-typed programs don't evaluate - that is, reach a value. This makes two-sided type systems suitable for incorrectness reasoning in higher-order program verification, which we illustrate through an application to precise data-flow typing in a language with constructors and pattern matching. Finally, we investigate the internalisation of the meta-level negation in the system as a complement operator on types. This motivates an alternative semantics for the typing judgement, which guarantees that ill-typed programs don't evaluate, but in which well-typed programs may yet go wrong.

Reliability Function and Boolean Ring

Reliability functions are fundamental mathematical models in engineering reliability theory, giving the probability that a system or component functions over a given time period. Boolean rings are algebraic structures with addition, multiplication, and complementation operations that follow certain axioms. This paper shows that collections of reliability functions form Boolean rings under natural definitions of the Boolean operators. Although reliability theory and Boolean rings are mature subjects, the formal connection between them has not been fully recognized. Establishing this link allows the extensive body of results from Boolean ring theory to be applied in analyzing reliability functions. The Boolean perspective leads to simplified calculations and new reliability theorems. We provide mathematical background on reliability functions and Boolean rings before presenting the formal proof that reliability functions satisfy Boolean ring properties. Several examples demonstrate how the Boolean viewpoint yields insight into reliability calculations, system structure functions, and other areas. The relationships revealed here unify reliability theory and Boolean algebra, while offering opportunities for further theoretical developments in both fields. This work elucidates the underlying Boolean structure inherent in reliability modeling.

Flexible resource dynamic aggregation regulation method of virtual power plant to ensure more renewable energy generation

The development of large-scale sustainable energy has affected the security of electricity systems. Virtual power plant (VPP) realize multi-energy synergistic complementation and efficient operation mainly with electricity. The paper proposed a flexible resource dynamic aggregation regulation method of VPP to ensure more renewable energy generation, which enables the efficient operation of multiple energy sources with electricity as the main source through collaborative and complementary measures. Firstly, the operating mode of VPP under market mechanisms is proposed, and the operating mechanism for VPP's participation in the market is established. Secondly, the types of flexible resources are identified, a cooperative game relationship between multiple autonomous agents within the power grid and VPP is formed to support an intelligent control method for VPP considering the complementarity of multiple energy sources and carbon emissions. Finally, a novel inverse cotangent compound differential evolution (NICCDE) algorithm is proposed by combining the innovative composite differential evolution algorithm with the tangent function, which realizes the flexible synergistic utilization of multiple energy sources. and reduces the training cost of the algorithm. Case studies from the conducted case study demonstrate the significant cost reduction achieved by the proposed method in comparison to conventional approaches when applied to VPP. Moreover, the utilization of this method enhances the precision and comprehensiveness of the search outcomes, thereby augmenting the overall effectiveness of the VPP system.

Nondeterministic operational complexity in subregular languages

We study the nondeterministic state complexity of basic regular operations on subregular language families. In particular, we focus on the classes of combinational, singleton, finite, finitely generated left ideal, symmetric definite, group, star, comet, two-sided comet, ordered, and power-separating languages, and consider the operations of intersection, union, concatenation, power, Kleene star, reversal, and complementation. We get the exact complexity in all cases, except for complementation of singleton languages where we only have a lower bound n and an upper bound n. The complexity of all operations on combinational languages is given by a constant function, except for the k-th power where it is k+1. For all considered operations, the known upper bounds for left ideals are met by finitely generated left ideal languages, and the known lower bound 2n−1 for complementation on left ideals is tight also for symmetric definite languages. The complexity of intersection on singleton languages is min⁡{m,n}, and for all other operations, except for complementation, the known upper bounds for finite languages are met by singleton languages. The nondeterministic state complexity of the k-th power, star, and reversal on star languages is n, and the nondeterministic state complexity of complementation on group languages is max⁡{n,(n−1⌊n/2⌋)}. In all the remaining cases, the nondeterministic state complexity of all considered operations is the same as in the regular case, although sometimes we need to use a larger alphabet to describe the corresponding witnesses.

Computation of Transmission Eigenvalues by the Regularized Schur Complement for the Boundary Integral Operators

This paper is devoted to the computation of transmission eigenvalues in the inverse acoustic scattering theory. This problem is first reformulated as a two by two boundary system of boundary integral equations. Next, utilizing the Schur complement technique, we develop a Schur complement operator with regularization to obtain a reduced system of boundary integral equations. The Nystrom discretization is then used to obtain an eigenvalue problem for a matrix. We employ the recursive integral method for the numerical computation of the matrix eigenvalue. Numerical results show that the proposed method is efficient and reduces computational costs.

A new method of multi‐attribute group decision making based on hesitant fuzzy soft expert information

AbstractInteresting insights into uncertain multi‐attribute decision‐making systems have come from the theory of hesitant fuzzy sets. Outputs become more reliable when a decision‐maker is not forced to produce one single assessment in the presence of hesitation. Hesitancy in fuzzy environments grants more flexibility to the decision‐maker, therefore enhancing the validity of all subsequent results. Due to these facts, hesitation has become a landmark in the enhancement of fuzzy‐soft‐inspired theories, and techniques using models such as hesitant fuzzy soft sets or hesitant (fuzzy) ‐soft sets have thrived. However, the literature lacks a comparable improvement for fuzzy soft expert information. The main goal of this paper is the integration of both hesitation and fuzziness with the characteristics of soft expert sets. The resulting hybrid model is therefore called hesitant fuzzy soft expert set, and its utilization for multiple attribute group decision‐making (MAGDM) will be investigated too. To that purpose, we first study the required algebraic operations and properties (subsethood and equality, complement, union, intersection, plus AND and OR operation). Respective numerical examples illustrate their utilization. We also establish some important laws for hesitant fuzzy soft expert sets, including commutativity, associativity, distributivity, and De Morgan's laws. Moreover, we define four novel types of level soft expert sets for fuzzy soft expert sets, and three types of fuzzy soft expert sets for hesitant fuzzy soft expert sets, which are: ‐level soft expert sets, level soft expert sets regarding fuzzy threshold, mid‐level soft expert sets and top‐level soft expert sets. Armed with these tools, we present a natural algorithm for MAGDM under hesitant fuzzy soft expert information that is accompanied by an illustrative application that concerns a selection problem for the samples of electric vehicles manufactured by different companies. This is followed by a comparison with existing mathematical tools such as hesitant fuzzy soft sets and fuzzy soft expert sets, in order to show the advantages of our work over them. In the end, we provide some future research directions.

Set algebra — based algebraic evolutionary algorithm for binary optimization problems

In order to design algebraic evolutionary algorithms by set algebra, the intersection, union, complement, difference and symmetric difference operations of 0-1 vectors on {0,1}n are firstly defined based on set operations. Then, the isomorphism between algebraic system defined on {0,1}n and set algebra defined on power set P(Ω) of set Ω is proved. Therefrom a simple and fast implement method of set algebra is proposed. Third, symmetric difference operator and asymmetric mutation operator are successively proposed based on set algebra, they have global exploration and local exploitation capabilities respectively. On this basis, a novel algebraic evolutionary algorithm, named set algebra-based heuristic algorithm (SAHA), is proposed based on the operations of 0-1 vectors on {0,1}n for solving binary optimization problems. For verifying the performance of SAHA, it is used to solve 0-1 knapsack problem (0-1KP) and knapsack problem with single continuous variable (KPC), respectively. The comparison with the state-of-the-art algorithms of solving 0-1KP and KPC shows that SAHA can not only obtain excellent calculation results, but also is faster speed, it is most competitive for solving binary optimization problems.

A multi-objective optimization model for the coordinated operation of hydropower and renewable energy

The rapid growth of wind and solar energy sources in recent years has brought challenges to power systems. One challenge is surging wind and solar electric generation, understanding how to consume such generation is important. Achieving the complementarity of hydropower and renewable energies such as wind and solar power by utilizing the flexible regulation performance of hydropower is helpful to provide firm power to help renewable energy consumption. However, the multi-energy complementary operation mode will change the traditional hydropower operation mode, causing challenges to the comprehensive utilization of hydropower. In this paper, a multi-objective optimal scheduling model is built by considering coordinated hydro-wind-solar system peak shaving and downstream navigation. First, the Gaussian mixture model is adopted to quantify the uncertainty of wind and solar power. Then, a hydro-wind-solar coordinated model was built to obtain the standard deviation of the residual load and the standard deviation of the downstream water level. Finally, the ε-constraint method is used to solve for the Pareto optimality. The results demonstrate the following: 1) The proposed model can effectively determine hydropower output schemes that can coordinate wind and solar power output to reconcile peak shaving and navigation; 2) The downstream hydropower stations’ reverse regulation of the upstream hydropower station is a positive factor in reconciling conflicts; and 3) Reasonable planning of wind power and solar power is helpful for hydro-wind solar power complement operation.

Vector Similarity Measures of Dual Hesitant Fuzzy Linguistic Term Sets and Their Applications

The dual hesitant fuzzy linguistic term set (DHFLTS) is defined by two functions that express the grade of membership and the grade of non-membership using a set of linguistic terms. In the present work, we first quote an example to point out that the existing complement operation of DHFLTS is on the wrong track. Meanwhile, we redefine this operation to fill the holes in the existing ones. Next, the notion of information energy under a dual hesitant fuzzy linguistic background is provided in order to build the criteria weight determination model. To further facilitate the theory of DHFLTS, we propose two vector similarity measures, i.e., Jaccard and Dice similarity measures, and their weighted forms for DHFLTS. In addition, we pioneer some generalized similarity measures of DHFLTSs and indicate that the Dice similarity measures are particular instances of the generalized similarity measures for some parameter values. Afterward, the similarity measures-based model with unknown weight information under the background of dual hesitant fuzzy linguistic environment is constructed. Lastly, an illustrated example is included to validate the method’s application, along with sensitivity analysis and comparative analysis, demonstrating the practicality and validity of its results.

Binary operations on neuromorphic hardware with application to linear algebraic operations and stochastic equations

Non-von Neumann computational hardware, based on neuron-inspired, non-linear elements connected via linear, weighted synapses—so-called neuromorphic systems—is a viable computational substrate. Since neuromorphic systems have been shown to use less power than CPUs for many applications, they are of potential use in autonomous systems such as robots, drones, and satellites, for which power resources are at a premium. The power used by neuromorphic systems is approximately proportional to the number of spiking events produced by neurons on-chip. However, typical information encoding on these chips is in the form of firing rates that unarily encode information. That is, the number of spikes generated by a neuron is meant to be proportional to an encoded value used in a computation or algorithm. Unary encoding is less efficient (produces more spikes) than binary encoding. For this reason, here we present neuromorphic computational mechanisms for implementing binary two’s complement operations. We use the mechanisms to construct a neuromorphic, binary matrix multiplication algorithm that may be used as a primitive for linear differential equation integration, deep networks, and other standard calculations. We also construct a random walk circuit and apply it in Brownian motion simulations. We study how both algorithms scale in circuit size and iteration time.

Four-Way Turiyam based Characterization of Non-Euclidean Geometry

Recently, a problem is addressed while dealing the data with Non-Euclidean Geometry and its characterization. The mathematician found negation of fifth postulates of Euclidean geometry easily and called as Non-Euclidean geometry. However Riemannian provided negation of second postulates also which still considered as Non-Euclidean. In this case the problem arises what will happen in case negation of other Euclid Postulates exists. Same time total total or partial negation of Euclid postulates fails as hybrid Geometry. It become more crucial in case the data is unknown, incomplete or exists beyond the three-way space as heteroclinic pattern. To understand this problem, the current paper tried to distinguish Euclidean, Non-Euclidean, Anti-Geometry, Neutrogeometry and Turiyam or Unknown geometry using the complement operator with an example.

Research on dynamic energy optimization strategy of park smart energy system with complementary multi-energy

The only way to develop a smart grid is to build a smart energy system in the park with coordinated optimization of source, network and load, and flexible interaction. In order to achieve the best operation of the multi-energy complementation of cooling, heating, and electricity in the smart energy system of the park when the output of renewable energy is uncertain, the Latin hyper square sampling method and the synchronous back generation method are used to analyze and model the uncertainty of renewable energy in the system. A model for the optimal operation of the integrated energy system for cooling, heating, and electricity is established. This model considers various demand response measures of controllable load and electric energy storage and considers how source, load, and storage can work together to reduce energy consumption. After conducting a multi-scenario simulation study, it is determined that this approach may significantly boost the park’s operational dependability and economic performance.

Construction of Fuzzy Linguistic Approximate Concept Lattice in an Incomplete Fuzzy Linguistic Formal Context

Uncertainty research is one of the critical problems in artificial intelligence. In an uncertain environment, a large quantity of information is expressed in linguistic values. Aiming at the missing linguistic-valued information, we first propose incomplete fuzzy linguistic formal context and then discuss the fuzzy linguistic approximate concept. Our proposal can describe the attributes of objects from two aspects simultaneously. One is an object's essential attributes, and another includes the essential and possible attributes. As a result, more object-related information can be obtained to reduce information loss effectively. We design a similarity metric for correcting the errors caused by the initial complement operation. We then construct a corresponding fuzzy linguistic approximate concept lattice for the task of approximate information retrieval. Finally, we illustrate the applicability and feasibility of the proposed approach with concrete examples, which clearly show that our approach can better deal with the linguistic-valued information in an uncertain environment.

An approach in medical diagnosis based on Z-numbers soft set.

In the process of medical diagnosis, a large amount of uncertain and inconsistent information is inevitably involved. There have been many fruitful results were investigated for medical diagnosis by utilizing different traditional uncertainty mathematical tools. It is found that there is limited study on measuring reliability of the information involved are rare, moreover, the existed methods cannot give the measuring reliability of every judgment to all symptoms in details. It is quite essential to recognize the impact on the reliability of the fuzzy information provided under inadequate experience, lack of knowledge and so on. In this paper, the notion of the Z-numbers soft set is proposed to handle the reliability of every judgment to all symptoms in details. The study in this paper is an interdisciplinary approach towards rapid and efficient medical diagnosis. An approach based on Z-numbers soft set (ZnSS)to medical diagnosis has been developed and is used to estimate whether two patterns or images are identical or approximately. The notion of Z-numbers soft set is proposed by combing the theory of soft set and Z-numbers theory. The basic properties of subset, equal, intersection, union and complement operations on the Z-numbers soft sets are defined and the similarity measure of two Z-numbers soft sets are also discussed in this paper. An illustrative example similar to existing studies is showed to verify the effectiveness and feasibility, which can highlight the proposed method and demonstrate the solution characteristics. Diagnosing diseases by uncertainty symptoms is not a direct and simple task at all. The approach based on ZnSS presented in this paper can not only measure reliability of the information involved, but also give the measuring reliability of every judgment to all symptoms in details.

A novel enhanced quantum image representation based on bit-planes for log-polar coordinates

Quantum image representation has a significant impact in quantum image processing. In this paper, a bit-plane representation for log-polar quantum images (BRLQI) is proposed, which utilizes $(n+4)$ or $(n+6)$ qubits to store and process a grayscale or RGB color image of $2^n$ pixels. Compared to a quantum log-polar image (QUALPI), the storage capacity of BRLQI improves 16 times. Moreover, several quantum operations based on BRLQI are proposed, including color information complement operation, bit-planes reversing operation, bit-planes translation operation and conditional exchange operations between bit-planes. Combining the above operations, we designed an image scrambling circuit suitable for the BRLQI model. Furthermore, comparison results of the scrambling circuits indicate that those operations based on BRLQI have a lower quantum cost than QUALPI. In addition, simulation experiments illustrate that the proposed scrambling algorithm is effective and efficient.

An Absolutely New Attempt to Vague Soft Bitopological Spaces Basis at Newly Defined Operations regarding Vague Soft Points

In this study, new operations of union, intersection, and complement are defined with the help of vague soft sets in a new way that is in both true and false statements, union is defined with maximum, and intersection is defined with minimum. On the basis of these operations, vague soft topology is defined. Pairwise vague soft open sets and pairwise vague soft closed sets are defined in vague soft bitopological structures (VSBTS). Moreover, generalized vague soft open sets are introduced in VSBTS concerning soft points of the space. On the basis of generalized vague soft open sets, separation axioms are also introduced. In continuation, these separations axioms are engaged with other important results in VSBTS.

E-Cyclist: Implementation of an Efficient Validation of FOLID Cyclic Induction Reasoning

Checking the soundness of cyclic induction reasoning for first-order logic with inductive definitions (FOLID) is decidable but the standard checking method is based on an exponential complement operation for B\"uchi automata. Recently, we introduced a polynomial checking method whose most expensive steps recall the comparisons done with multiset path orderings. We describe the implementation of our method in the Cyclist prover. Referred to as E-Cyclist, it successfully checked all the proofs included in the original distribution of Cyclist. Heuristics have been devised to automatically define, from the analysis of the proof derivations, the trace-based ordering measures that guarantee the soundness property.

Interval quadripartitioned Neutrosophic Sets

Quadripartitioned neutrosophic set is a mathematical tool, which is the extension of neutrosophic set and n-valued neutrosophic refined logic for dealing with real-life problems. A generalization of the notion of quadripartitioned neutrosophic set is introduced. The new notion is called Interval Quadripartitioned Neutrosophic set (IQNS). Quadripartitioned neutrosophic set is developed by combining the Quadripartitioned neutrosophic set and interval neutrosophic set. Several set theoretic operations of IQNSs, namely, inclusion, complement, and intersection are defined. Various properties of set-theoretic operators of IQNS are established.