Crisp Numbers Research Articles

Розв’язання задачі комівояжера на основі методу відпалу з урахуванням нечіткості сприйняття плину часу

This paper investigates the use of fuzzy numbers and the annealing method to improve the results of the traveling salesman problem (TSP) by more accurately representing real-world circumstances, where the value of the objective function represents the subjective perception of the length of the time interval required to travel between cities. TSP is a classic combinatorial optimization problem that involves finding the shortest route between a set of cities. Fuzzy numbers are used to model input inaccuracy and uncertainty, as they allow for a more detailed representation of real-world constraints and factors that may affect the problem. The annealing method is used to optimize the TSP solution by gradually decreasing the temperature of the system, which allows exploring different solutions and avoiding getting stuck in local minima. To demonstrate the effectiveness of this approach, a Python program was developed that was used to compare the results of the TSP problem using crisp and fuzzy numbers using the annealing method. The results show that the use of fuzzy numbers, particularly triangular and parabolic, with the annealing method leads to a significant improvement in the results of the TSP problem compared to the use of crisp numbers, assuming a model is called realistic if it has possible deviations from the expected fixed mean. Computational results of the program are presented and analyzed, demonstrating the potential of this approach for real-world optimization problems involving imprecise or uncertain data and which can be particularly applied to the optimization of processes with subjective time perception.

Green winner determination method based on environmental performance and minimum adjustment consensus in 4PL transportation service procurement.

Due to the intensified environmental protection consciousness of enterprises and consumers, the green winner determination (GWD) considering environmental performance becomes very important for the 4PL transportation service procurement. In this paper, a new GWD method is studied, which considers different types of attributes including those related to environmental performance and the consensus reaching process (CRP). To characterize multiple types of attributes, linguistic terms, interval numbers, and crisp numbers are combined. To achieve an acceptable consensus level among linguistic evaluations given by different experts, a minimum adjustment consensus model is constructed. And on this basis, an interactive CRP is proposed. Integrating the heterogeneous information addressing process and the CRP, a HC-VIKOR method is developed to promote the 4PL's operational efficiency and service quality. Further, a numerical example is designed to demonstrate the effectiveness of the proposed method. Sensitivity analysis reveals that both the acceptable consensus threshold and the weight of group utility have a significant influence on the winner determination result. Comparison analysis shows that the proposed method outperforms the existing methods. Our study not only extends the traditional winner determination but also provides decision support for the 4PL to provide transportation services efficiently.

Highly efficient numerical scheme for solving fuzzy system of linear and non-linear equations with application in differential equations

ABSTRACT In this research, we suggested a numerical iterative scheme for investigating the numerical solution of fuzzy linear and nonlinear systems of equations, particularly where the linear or nonlinear system co-efficient is a crisp number and the right-hand side vector is a triangular fuzzy number. Triangular fuzzy systems of linear and nonlinear equations play a critical role in a variety of engineering, scientific challenges, mathematics, chemistry, physics, artificial intelligence, biology, medical, economics, finance, statistics, machine and deep learning, computer science, robotics and smart cars, programming, in the military and engineering industries, linear and nonlinear programming problems and traffic flow problems. In biomedical engineering, fluid flow problems, and differential equations, triangular fuzzy linear and nonlinear systems of equations also play a key role in determining the level of uncertainty. Convergence analysis illustrates that the proposed numerical technique's order of convergence for solving a triangular fuzzy system of linear and nonlinear equations is three. The newly developed numerical scheme was then applied to solve several triangular fuzzy boundary value problems. In terms of convergence rate, computing time, and residual error, numerical test problems indicate that the newly developed methods are more efficient than the current methods in the literature.

Type-2 intuitionistic fuzzy TODIM for intelligent decision-making under uncertainty and hesitancy

The classical TODIM considers the crisp numbers to handle the information. However, in a real-world applicative context, this information is bounded by noise and vagueness and hence uncertain. There are wide range of works in the literature which utilizes fuzzy sets to handle the uncertainty in the various dimensions. However, there is a constraint of hesitancy in such decision-making problems due to the involvement of various decision-makers. Also, in the TODIM method, decision-maker’s bounded rationality and psychological behavior are also taken into consideration which adds up the hesitation and considers the problem with higher dimension of uncertainty. There are various applications of fuzzy TODIM using type-2 fuzzy numbers where uncertainty is being handled in more than one dimensions and also the introduction of intuitionistic fuzzy numbers where hesitancy factor of a decision-maker is taken into account. This paper targets to handle the uncertainty in more than one dimension keeping the hesitancy part into consideration for intelligent decision-making. Therefore, a novel trapezoidal type-2 intuitionistic fuzzy set (TrT2 IFS) is proposed, which is an aggregation of several triangular IFSs having upper and lower membership along with a non-membership value. For this TrT2 IFS, we have defined the generation procedure, operations, comparison and distance between such TrT2 IFSs. In addition, the decision-maker weights and criterion weight computation are also presented with respect to the TODIM approach. Furthermore, we have applied this extended TrT2 IFS-based TODIM method into a renewable energy resource selection problem of multi-criteria decision-making.

Assessment and ranking of occupational health and safety risks in the machinery maintenance unit of dam construction projects using the combined approach of methods (Fmea and Fuzzy Topsis)

A dam construction project is exposed to various risks due to the amount of investment, duration of implementation, the complexity of design and implementation, etc., and these risks will lead to human and financial losses. Thus, based on three principles, including (i) time, (ii) quality, and (iii) cost, identifying, assessing, ranking risks, and Occupational Safety and Health (OSH) can have a significant impact on project objectives. The machinery maintenance unit has the largest share of the organization's cost model. Manpower, tools, spare parts, and working standards of planning and scheduling are the most important elements of the implementation of the maintenance process in these organizations. Methodology In the present research, first, all the failure modes of the machinery maintenance unit were identified. Then, to assess and rank them, in addition to the three traditional attributes in Failure Modes and Effects Analysis (FMEA) (severity-S, occurrence-O, and detection-D) and calculate the Risk Priority Number (RPN), we used fuzzy numbers corresponding to real-world problems instead of crisp numbers for risk ranking.

A new hybrid risk assessment method based on Fine-Kinney and ANFIS methods for evaluation spatial risks in nursing homes

Today, as the elderly population in the world increases, the increase in those living in nursing homes causes their problems to be even more important. Spatial hazards cause injury and death most of the time, therefore should be evaluated risks then corrective and preventive actions should be implemented. Fine-Kinney is one of the most widely used risk assessment methods, but it has some shortcomings. One of them is that risk factors such as probability, frequency, and severity are accepted as equally important, but they can have different importance weights in real-life applications. Another is that experts assess the risk magnitudes using their opinions, who usually tend to use linguistic expressions instead of crisp numbers, in incomplete information and uncertain situations. The last is that the experts' experiences are not effectively incorporated into the automation of the risk assessment. The adaptive neuro-fuzzy inference system (ANFIS) method, which is a machine learning method, can overcome all these shortcomings.In this study, a novel hybrid risk assessment method based on Fine-Kinney and ANFIS is developed to predict the class of a new occurring risk. The hybrid method was applied to nursing homes located in Turkey. The risk classes predicted with the hybrid method were compared to ones found in the traditional Fine-Kinney method. It was determined that the prediction accuracy and Fleiss kappa value of the new hybrid method were 95.745% and 0.929 respectively. Thus, the hybrid method can be used instead of the traditional Fine-Kinney method to determine the class of a new risk, because it does not require a large number of experts and provides a faster assessment.

An improved risk prioritization method for propulsion system based on heterogeneous information and PageRank algorithm

Failure mode and effect analysis (FMEA) has been widely applied as a powerful reliability analysis technique to identify and eliminate system failures in various industries. Though many contributions have been made to improve the traditional FMEA, some challenges still exist. For example, how to deal with different types of risk information involved in FMEA due to the heterogeneity of risk factors. Additionally, the influence of the relationship between failure modes and the attenuation effect on the final risk priority deserves further study. Motivated by these challenges, we propose an improved risk prioritization method. First, heterogeneous information expression structures (i.e., crisp numbers, interval numbers, triangular fuzzy numbers, linguistic Z-numbers) are utilized to describe risk factors from quantitative and qualitative aspects. Then, the initial risk priority numbers (RPNs) of failure modes are determined without transforming different representation structures into a unified one. Next, the total relation influence matrix is generated considering attenuation effects. Afterward, the PageRank algorithm is extended to obtain the final risk priority by analyzing the mutual influence between failure modes and their initial RPNs simultaneously. Finally, a case study of FMEA for a shipboard medium voltage DC power system is presented to verify the effectiveness of the proposed method. The results indicate that our method can process the heterogeneous risk information more flexibly and prioritize the failure modes more reasonably.

A fairness-concern-based LINMAP method for heterogeneous multi-criteria group decision making with hesitant fuzzy linguistic truth degrees

Heterogeneous multi-criteria group decision making (MCGDM) is a hot topic in the decision analysis field. This paper proposes a fairness-concern-based LINMAP (Linear Programming Technique for Multidimensional Analysis of Preference) method for heterogeneous MCGDM with hesitant fuzzy linguistic (HFL) truth degrees. Heterogeneous evaluation information includes crisp numbers, interval numbers, intuitionistic fuzzy values (IFVs), trapezoidal fuzzy numbers (TrFNs) and hesitant fuzzy sets (HFSs). This paper introduces the fairness concern to calculate the HFL consistency and the HFL inconsistency indices. Based on the framework of LINMAP, a bi-objective HFL programming model is built to derive the criteria weights, the positive ideal fairness vector (PIFV) and the negative ideal fairness vector (NIFV) for each decision maker (DM) simultaneously. Based on the TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution), a multi-objective programming model is built to obtain DMs’ weights. The alternatives ranking is derived by comprehensive collective relative closeness degrees. Finally, a real example is applied to verify effectiveness and superiority of this heterogeneous MCGDM method. The proposed heterogeneous MCGDM method provides a very useful approach for MCGDM with heterogeneous information.

Evaluation of the Environmental Effect of Automated Vehicles Based on IVIULWG Operator Development

The automated vehicle (AV) industry is a new growing industry of great interest. The environmental friendliness of AVs represents a core characteristic of their leap-forward development. The environmental effect of AVs, including its evaluation framework and algorithm, is a leading research area for continued technological innovation and market development in this field. This study focuses on three environmental aspects: the energy effect of AVs, the traffic effect of AVs on ground space, and the air effect of AVs on three-dimensional atmospheric space. First, an environmental effect evaluation indicator system that includes nine indices for AVs is constructed to be the basis for management decision making. Second, the interval-valued intuitionistic uncertain fuzzy (IVIUF) evaluation variable is used to solve the problem of data features that incorporate “crisp numbers” and “fuzzy numbers”. Moreover, geometric weights are added based on the generalized correlation aggregation operator. Then, the IVIUL-weighted geometric (IVIULWG) operator is developed, which includes the objective and subjective information of “crisp numbers” and “fuzzy numbers” and makes the mathematical characteristics more scientific and accurate. Finally, an evaluation example is used to validate the effectiveness and practicability of the algorithm.

BULANIK CPM İLE DOĞRUSAL PROGRAMLAMA: İNŞAAT ENDÜSTRİSİNDE BİR VAKA ÇALIŞMASI

İnşaat sektöründe karmaşık projelerin planlanması ve kontrolü için Kritik Yol Metodunun (KYM), yaygın olarak kullanılan yararlı bir araç olduğu kanıtlanmıştır. Ancak, faaliyetlerin süresi net sayılarla temsil edildiğinden, kritik yol analizinde faaliyet sürelerini kesin olarak tahmin etmek zorlaşır. Buna ek olarak, vakaların çoğu, görev sürelerinin sübjektif olarak hesaplanmasını gerektirir ve bu da faaliyetlerin süresi hakkında belirsizliğe neden olur. Bu makale, bulanık kümelere dayalı bir yaklaşım önererek bu sorunları ele almaktadır. Bu çalışmanın amacı, bir inşaat projesinin kritik yolunu ve tamamlanma süresini hesaplamak için bulanık sayıların nasıl kullanılacağını göstermektir. Çalışma kapsamında proje faaliyetlerine üçgen bulanık süreler verilerek iki ayrı çözüm algoritması oluşturulmuştur. Örnek bir inşaat projesinin kritik yolu ve proje süresi ilk çözüm algoritmasında proje faaliyetlerine üçgen bulanık süreler atanarak ve doğrusal programlama modeli kullanılarak hesaplanırken, ikinci çözüm algoritmasında bulanık proje süresi ve kritik yol Alfa kesme yöntemi (α -Kesme Yöntemi) kullanılarak hesaplanmış ve daha sonra Centroid yöntemi (Alanların Merkezi Yöntemi) kullanılarak netleştirilmiştir. Bu proje için kritik yol ve tamamlanma süresi önceden bilindiğinden, iki çözüm algoritması karşılaştırılmıştır. Kritik yol yöntemi yerine Bulanık kritik yol yönteminin kullanılmasının bu çalışmanın önemini vurgulayacağı umulmaktadır.

Extended Transportation Models Based on Picture Fuzzy Sets

Picture fuzzy set (PFS) is an extension of intuitionistic fuzzy set, and it is capable to analyze the transportation problems that contain uncertain and vague information. PFSs are applicable in situations where human decisions demand more variety of responses: yes, abstain, no, and rejection which cannot be addressed in the traditional FSs and IFSs. In this paper, we propose a technique to solve fully picture fuzzy transportation problems (FPFTPs) which are based on picture fuzzy linear programming formulation. Fully picture fuzzy transportation problems are developed by considering all the variables and parameters as nonnegative trapezoidal picture fuzzy numbers. A ranking function is used to transform picture fuzzy numbers into crisp numbers. A model is presented to explain the suggested scheme. Finally, comparison analysis of fully picture fuzzy transportation model with fully intuitionistic fuzzy transportation model and fully fuzzy transportation model is presented with pictorial illustrations.

MODEL LOGISTIK FUZZY DENGAN ADANYA PEMANENAN PROPORSIONAL

The logistic growth model with proportional harvesting is a population growth model that takes into account harvesting factors. In real life, not all conditions can be known with certainty, such as different growth rates in each population and harvest rates depending on the needs of the harvester. To overcome these conditions, there is a concept that accommodates the problem of uncertainty, namely the fuzzy concept. This concept can be induced into a logistic model with proportional harvesting which assumes the intrinsic growth rate and the harvest rate is expressed by fuzzy numbers. The purpose of this research is to form a logistic model with fuzzy proportional harvesting, analyze the stability of the model, and form a numerical simulation. This study uses the alpha-cut method to generalize the intrinsic growth rate and harvest rate from crisp numbers to fuzzy numbers, then the Graded Mean Integration Representation (GMIR) method to defuzzify the model, and the linearization method to analyze the stability of the model. The results of this study obtained a logistic model with proportional harvesting. Then the model was developed into a logistic model with fuzzy proportional harvesting by assuming the intrinsic growth rate and the harvest rate expressed by fuzzy numbers. From the model obtained 2 equilibrium points, namely the first equilibrium point is unstable and the second equilibrium point is asymptotically stable under certain conditions. Model simulation is given to show illustration of stability analysis. From the simulation, it can also be shown that the higher the graded mean value, the lower the population.

Selection of Supply Chain Sustainability Management System by Fuzzy Additive Preference Programming Method

A selection of suitable sustainability management systems (SMS) is a major part of supply chain strategies to create a competitive advantage, reduce total costs, and manage long-term sustainability. A framework and method for prioritizing supply chain SMSs are presented in this research. Analytic hierarchy process (AHP) is the most common method for alternative selection in multi-criteria decision-making (MCDM). However, complex information is mixed with ambiguity and uncertainty, which makes decision makers unable to use precise or crisp numbers, so fuzzy numbers are presented to remedy this difficulty. Therefore, this research proposes a fuzzy additive preference programming (FAPP) to select the optimum SMS for a supply chain. FAPP method can produce the unique normalized optimal priority vector of fuzzy pairwise comparison matrices for SMS selection effectively with linear constraints. The additive linear constraints can eliminate the weaknesses of existing methods and equalize the upper and lower triangular fuzzy judgments. In addition, the proposed method can identify abnormal pairs of fuzzy judgments that cause inconsistency. The proposed methodology can prioritize the key criteria which lead to the selection of the most appropriate SMS. An example of SMS selection in a rubber factory demonstrates the feasibility and validity of the proposed method.

Fully Bipolar Single-Valued Neutrosophic Transportation Problems

Transportation problem (TP) has its uses in real life because it has versatile applications. Real-life problems are often uncertain due to which it is difficult to find the accurate cost. The fuzzy set and intuitionistic fuzzy set are useful for handling the uncertainty, but these also have some limitations. For that reason, in this study, we worked on another set of values called bipolar single-valued neutrosophic set (BSNS) which is the generalization of crisp sets, fuzzy sets, and intuitionistic fuzzy sets to handle the uncertain, unpredictable, and insufficient information in real-life problems. In this study, we develop a new technique for solving transportation problems based on bipolar single-valued neutrosophic sets having nonnegative triangular bipolar single-valued neutrosophic numbers (TBSNNs). A score function is used to transform bipolar single-valued neutrosophic numbers (BSNNs) into crisp numbers. We compare our proposed model with fuzzy transportation and intuitionistic fuzzy transportation models and proved that bipolar single-valued neutrosophic transportation model is more admirable than the existing models. Furthermore, we apply the proposed technique to fully solve the bipolar single-valued neutrosophic transportation (FBSNT) model.

Fuzzy integrated two stage vendor - buyer inventory model with crisp number

Fuzzy integrated two stage vendor - buyer inventory model with crisp number

A dynamic case-based reasoning system for responding to infectious disease outbreaks

Infectious diseases are a global public health problem, which requires timely and effective responses. This study proposes a novel model that contributes to the development of such responses. First, the problem scenario features of infectious disease emergency scenarios are extracted, and the problem scenario is structurally described. A Markov model is adopted to analyze the scenario evolution of the infectious disease outbreaks. Then, a dynamic case-based reasoning model is built. Different matching algorithms are designed for crisp symbols, crisp numbers, interval numbers, and fuzzy linguistic variables. The similarity between the target scenario and various historical scenarios is calculated. Finally, an optimized dynamic emergency decision guide is provided. An experiment is conducted to test the validity and feasibility of the proposed method. The results suggest that the model can realistically simulate the process of infectious disease outbreaks and quickly match the recorded scenarios to generate effective and real-time responses.

Knowledge Representation and Reasoning with an Extended Dynamic Uncertain Causality Graph under the Pythagorean Uncertain Linguistic Environment

A dynamic uncertain causality graph (DUCG) is a probabilistic graphical model for knowledge representation and reasoning, which has been widely used in many areas, such as probabilistic safety assessment, medical diagnosis, and fault diagnosis. However, the convention DUCG model fails to model experts’ knowledge precisely because knowledge parameters were crisp numbers or fuzzy numbers. In reality, domain experts tend to use linguistic terms to express their judgements due to professional limitations and information deficiency. To overcome the shortcomings of DUCGs, this article proposes a new type of DUCG model by integrating Pythagorean uncertain linguistic sets (PULSs) and the evaluation based on the distance from average solution (EDAS) method. In particular, experts express knowledge parameters in the form of the PULSs, which can depict the uncertainty and vagueness of expert knowledge. Furthermore, this model gathers the evaluations of experts on knowledge parameters and handles conflicting opinions among them. Moreover, a reasoning algorithm based on the EDAS method is proposed to improve the reliability and intelligence of expert systems. Lastly, an industrial example concerning the root cause analysis of abnormal aluminum electrolysis cell condition is provided to demonstrate the proposed DUCG model.

A heterogeneous multi-attribute case retrieval method for emergency decision making based on bidirectional projection and TODIM

Emergency decision making (EDM) entails a heterogeneous multi-attributes problem concerning crisp number, interval number, triangular fuzzy number, and intuitionistic fuzzy number, etc. However, it creates difficulties for the decision makers (DMs) in undertaking an optimal decision. The study focuses on the heterogeneous multi-attribute case retrieval method, which is an effective method to assist DMs in promptly generating an emergency alternative. Nevertheless, the accuracy and comprehensiveness of similarity measurement remains one of the major issues to be solved when obtaining the similar case set; this is a key step in EDM. A heterogeneous multi-attribute case similarity measurement based on bidirectional projection is firstly proposed to calculate the case similarities between the historical cases and the target case, which simultaneously considers distance and direction, along with the optimal model based on maximum entropy method and the nearness degree is constructed to determine the attribute weights. Furthermore, the DMs’ psychological behavior is considered during the selection process of the most suitable historical case via the TODIM (an acronym in Portuguese of interactive and multi-attribute decision making) method. Hence, an alternative of the most suitable historical case can be exploited to generate the alternative of target case. Finally, the proposed method is illustrated using a typhoon disaster, and its feasibility, effectiveness, and practicality are validated via sensitivity analysis and its comparisons with other methods.

Modified Ranking Function to Compute Fuzzy Matrix Games

Game theory problems (GTP) frequently occur in Economy, Business Studies, Sociology, Political Science, Military Activities, and so on are some of the subjects covered. To tackle the uncertainty in Games, the analysis of games in which the payoffs are represented by fuzzy numbers (FN) will benefit from fuzzy set theory (FST). The purpose of this paper is to develop an efficient technique for solving constraint matrix games (MG) with payoff trapezoidal fuzzy numbers (TFN). The description of the new ranking method is introduced for a constrained matrix with TFN and values. Stock market forecasting has been one of the most important research areas for decades. Stock market values are volatile, non-linear, complicated and chaotic. Based on a ranking function (RF), we used a new algorithm to solve the fuzzy game problem (FGP) employing TFN and also to try to get a desirable gain. Centered on the latest proposed ranking algorithm, the Fuzzy decision method is designed to analyze possible stock opportunists. The paper considers a zero-sum game for two persons in which TFN are fuzzy payoffs. A ranking method (RM) is proposed to convert TFN into crisp numbers (CN) and it is used to solve FGP. The fuzzy game (FG) issue with concept strategies pure minimax maximin is presented. This problem is converted into the crisp problem by a new RF and then solved using the arithmetic (oddment) method. With the help of numerical examples, the suggested technique is explained. This paper finalizes the conclusion and includes an outlook for future study in this direction

Fuzzy Multi-Objective Linear Programming Problem Using DM's Perspective

In this paper, a two-stage method has been proposed for solving Fuzzy Multi-objective Linear Programming Problem (FMOLPP) with Interval Type-2 Triangular Fuzzy Numbers (IT2TFNs) as its coefficients. In the first stage of problem solving, the imprecise nature of the problem has been handled. All technological coefficients given by IT2TFNs are first converted to a closed interval and then the objectives are made crisp by reducing a closed interval into a crisp number and constraints are made crisp by using the concept of acceptability index. The amount by which a specific constraint can be relaxed is decided by the decision maker and thus the problem reduces to a crisp multi-objective linear programming problem (MOLPP). In the second stage of problem solving, the multi-objective nature of the problem is handled by using fuzzy mathematical programming approach. In order to explain the methodology, two numerical examples of the proposed methodology in Production planning and Diet planning problems have also been worked out in this paper.