Crisp Numbers Research Articles

ABOUT OF THE ANNEALING METHOD USING FOR THE TRAVELING SALESMAN PROBLEM SOLUTION WITH THE FUZZY TIME PERCEPTION

Context. The article considers a technique for the use of fuzzy numbers and the annealing method for solving the traveling salesman problem, which is formulated as the problem of finding a route to visit a given number of cities without repetitions with a minimum duration of movement. The task of formalizing the algorithm for solving the traveling salesman problem by the annealing method using fuzzy numbers for subjective time perception is posed. The use of fuzzy numbers to increase the accuracy to represent real-world circumstances is proposed. Objective. The goal of the work is to develop an algorithm for solving the traveling salesman problem based on the implementation of the annealing method with fuzzy numbers representing the subjective time perception for traveling between the cities with the minimum perceived duration of movement along the route. Method. This paper proposes a method for solving the traveling salesman problem by the annealing method with fuzzy numbers for subjective time perception. A scheme for formalizing the procedure for solving the traveling salesman problem with the minimal perceived duration of movement along the route is described. A variant of the original traveling salesman problem is proposed, which consists in using fuzzy numbers to represent the uncertainty and subjective time perception in traveling between cities as opposed to regular crisp numbers to show regular distance and/or time of traveling. The results of the proposed algorithm for calculating solutions to the traveling salesman problem with minimization of the perceived duration of movement are presented, the obtained solutions are compared with the solutions found by other heuristic methods. Results. The method for solving the traveling salesman problem using the annealing method with fuzzy numbers for subjective time perception is developed. A variant of the original traveling salesman problem is proposed, which consists in using fuzzy numbers to represent the uncertainty and subjective time perception in traveling between cities as opposed to regular crisp numbers to show regular distance and/or time of traveling. The application of fuzzy numbers makes it possible to perform calculation over possibly uncertain or subjective data, making results more accurate in the case of realistic deviations from the expected mean values in distance coverage. The results of the proposed algorithm for calculating solutions to the traveling salesman problem with minimization of the perceived duration of movement are presented, the obtained solutions are compared with the solutions found by other heuristic methods. Conclusions. The paper considers a method for formalizing the algorithm for solving the traveling salesman problem using fuzzy numbers for subjective time perception. The use of fuzzy numbers to increase the accuracy to represent real-world circumstances is proposed. The scheme for formalizing the procedure for solving the traveling salesman problem with the minimal perceived duration of movement along the route is described. A variant of the original traveling salesman problem is proposed, which consists in using fuzzy numbers to represent the uncertainty and subjective time perception in traveling between cities as opposed to regular crisp numbers to show regular distance and/or time of traveling.

Neutrosophic fuzzy decision-making framework for site selection

These days, the area of interest based on canteen location selection in the university campus is a prominent topic. This research work directly affects students' convenience, accessibility and the total campus experience. An ideal canteen site not only provides good food to the students and staff but also takes care of their health by providing a healthy environment. In this paper, we choose various criteria and sub-criteria to calculate the best site for a canteen in the university campus. Location, space and layout, infrastructure and Utilities, safety and compliance and environmental factors as criteria with their corresponding sub-criteria are chosen for this work. Here, the data sets used are provided by decision makers in linguistic words and then converted into crisp numbers. The criteria and sub-criteria weights are considered by a popular multi criteria decision-making (MCDM) method named Criteria Importance Through Intercriteria Correlation, i.e., CRITIC method. Therefore, the ideal location for the canteen in the university campus was determined by applying the complex proportional assessment method (COPRAS), which is basically a ranking method of MCDM. And finally, sensitivity analysis was conducted to make sure the outcomes were flexible and stable.

Site Selection for Girls Hostel in a University Campus by MCDM based Strategy

The area of interest based on girls’ hostel site selection is an important topic nowadays. The research directly impacts the well-being, safety, and academic success of female students who are taking the facilities for accommodation while pursuing their academic courses. An efficient hostel location not only supports the physical safety of students but also takes care of their psychological and emotional well-being by providing a suitable and helpful living atmosphere. Different valuable criteria and sub-criteria are considered for evaluating the most suitable site for a girls' hostel. The security and protection, proximity to academic buildings, accessibility, health and safety, and environmental factors as criteria with their associated sub-criteria are considered for this study. Two decision makers are given data sets in linguistic terms and further converted into crisp numbers. The weights of the criteria and sub-criteria are determined by a well-known multi-criteria decision-making (MCDM) technique called the Entropy weighted method. Further, the most prioritized location for the girls' hostel on the university campus was evacuated using the weighted aggregated sum product assessment (WASPAS) based ranking MCDM method. Additionally, the sensitivity analysis was conducted to check the stability and flexibility of the results.

On the Construction of Admissible Orders for Tuples and Its Application to Imprecise Risk Matrices

The imprecision inherent in human opinions is not properly modeled by crisp numbers. Other more complex structures like intervals or tuples capture better the imprecision of human assessments. This makes them very useful in decision problems. However, they cannot be easily compared. Despite they grasp better decision-makers inaccuracy, the lack of a natural total order for such structures makes the determination of the best alternative a difficult task. In this contribution, we explore how to obtain new total orders for (ordered) tuples paying special attention to admissible orders (total orders that extend the lattice order). The resulting orders are applied to four-dimensional ordered tuples that represent risk assessments in an imprecise environment. In addition, two case studies involving risk matrices in educational transport and the construction of a metro station are also provided.

Hybrid Evaluation Model to assess the overall barriers of public-private partnership contracts

Public-private partnerships are increasingly being used in many countries as a common management technique to enhance the delivery of public services and major public infrastructure projects. Understanding the project characteristics and obstacles to a public-private partnership is key for those involved to improve implementation. This study focuses on barriers that arise from public-private partnership contact in field wastewater treatment plants and offers a methodology for evaluating the overall barrier for public-private partnership projects by analyzing the relationships between the barrier elements. A primary list of barrier factors and their eight latent project characteristics were collected through literature reviews, then using the Delphi technique to filter the barrier insights concerning project characteristics and develop the final list using field interviews. 207 valid questionnaires were obtained. The data was deconvoluted into crisp numbers using fuzzy set theory, and a second-order structural equation model was used to validate the conceptual model. The results of the study indicate that the availability payment capability group has the highest score of all the groups in this study. This study proposed a model that enables the private sector to measure and evaluate aggregate barriers in projects so that it is easier for decision-makers to determine whether they can choose the project or not.

Solving Fuzzy Assignment Problem of Pythagorean Fuzzy Numbers Exploiting a Software Tool

A fresh approach to computing assignment issues in the pythagorean fuzzy domain is the focus of this body of work. Pythagorean fuzzy numbers are utilised in this work for the sake of analysing working hours and working costs. For converting fuzzy sets integers into a crisp number, a new ranking process is used. Optimal solutions have to be found by using software techniques for minimization, maximization and unbalanced assignment problems. In the end, numerical examples are being solved by using the proposed technique.

A VIKOR Method Based on the Relative Closeness on Interval Linguistic Neutrosophic Uncertain Linguistic Numbers

In actual multi-attribute group decision-making problems, due to the complexity and uncertainty of objective things and the ambiguity of human thinking, decision-makers find it hard to give accurate evaluation information by crisp numbers. Even the weights of attribute values and decision-makers are ambiguous. At this time, we are more inclined to adopt the intuitive form of linguistic variables such as "excellent", "good" or "bad" to describe attribute values and weights. So in this paper, based on a neutrosophic set (NS), we further propose interval linguistic neutrosophic uncertain linguistic number (ILNULN). ILNULN combines interval linguistic neutrosophic and uncertain linguistic numbers, and it has the advantages of both. At the same time, due to the weighted arithmetic Bonferroni mean operator considering the interrelationship between aggregation parameters, therefore we combine the ILNULN and weighted arithmetic Bonferroni mean operator to propose the interval linguistic neutrosophic uncertain linguistic weighted arithmetic Bonferroni mean (ILNULWABM) operator. Finally, under the environment of interval linguistic neutrosophic and uncertain linguistic numbers, this article uses the linguistic weights and ILNULWABM operator to make VIKOR decision based on the relative closeness, and gives a practical example.

Формирование оптимального кредитного портфеля с использованием нечётко-множественного подхода

The problem of profit maximization from lending to a financial organization is considered. A set of loan con-tracts forms a loan portfolio, which makes it possible to use the portfolio theory in the study of the designated problem. We will consider the formation of a loan portfolio as the formation of an investment portfolio. The problem arises of redistributing the funds of the investment portfolio in order to minimize its possible risk and maximize profitability. This problem in classical formulation can be solved according to Markowitz risk diversification. Thus, the problem of risk minimization and profitability maximization from lending to a financial institu-tion can be represented as a fuzzy mathematical programming problem, and more specifically, as a fuzzy linear programming problem. The resulting problem is solved by considering discrete α-levels and is reduced to a set of linear programming problems constructed for each of them. A clear expected optimal solution can be obtained by applying the defuzzification method, which allows you to convert a fuzzy set into a crisp number.

Risk assessment of crane operation hazards using modified FMEA approach with Z-number and set pair analysis

In view of two key deficiencies of failure mode and effects analysis (FMEA) in crane operation hazard risk assessment—information loss and hazard grade lack, a crane operation hazard risk assessment model based on the Z-number and set pair analysis (Z-SPA) is proposed. In this research, the Z-number is used to address the information uncertainty. To reduce the risk losing evaluation information in the conversion of the Z-number into a crisp number, it is translated into an interval number. In this assessment model, the grades of crane operation hazards are determined using grade discrimination rules and a score function of a connection number based on set pair analysis. Two case studies are presented to exemplify the effectiveness of the approach. The first case validates the proposed method, and the second case illustrates the application of the modified FMEA to crane operation hazard risk analysis. Furthermore, the assessment results are analyzed and compared with previous studies to highlight the superiority of the proposed model.

About defuzzification methods influence on fuzzy traveling salesman problem’s solving

The article investigates the approach to using fuzzy numbers and the method of dynamic programming to find solutions to the traveling salesman problem, considering the fuzzy representation of time in real travel conditions. This allows for formulating a fuzzy optimization problem to find the best value of the objective function, which is determined by the amount of time required to travel between cities. The traveling salesman problem (TSP) is a classic problem of combinatorial optimization, which involves finding the shortest or fastest route among a set of cities. Fuzzy numbers are used to formalize the uncertainty and imprecision of input data, associated with the subjectivity in estimates of the duration of necessary travel intervals. For operating with fuzzy numbers, their transformation into a special form is proposed, and the formalization of the obtained fuzzy results into a crisp representation is carried out based on the center of gravity (CoG) method. A comparison of the results obtained based on solving the deterministic traveling salesman problem using defuzzified time distances and the defuzzification of the solution to the fuzzy traveling salesman problem was conducted. The results confirmed the dependency of the solution on the method of defuzzification. A program was developed that was used to compare the results of the traveling salesman problem using crisp and fuzzy numbers based on the dynamic method. A conclusion is drawn, indicating that the use of trapezoidal fuzzy numbers with the dynamic programming method leads to improved results of the problem compared to using crisp numbers based on the defuzzification of fuzzy distances. Methods of implementation and problematic areas of application of the computation results are presented and analyzed, demonstrating the constructiveness of the proposed approach for studying real processes.

A Type-2 Fuzzy u-Control Chart Considering Probability-Based Average Run Length

Fuzzy sets are an emerging trend in shaping the development of control charts for statistical process control. The sets are germane to vague data that comes from incomplete or inaccurate measurements. Nevertheless, fuzzy sets are inadequate in some areas of industry since their membership functions are crisp numbers. The fuzzy sets are not fully able to compute higher levels of uncertainty, which might degrade the performance of the analysis. Therefore, type-2 fuzzy sets are proposed to be merged with control charts since these sets are hypothesized to be more capable of detecting a defect in process control. This paper aims to develop interval type-2 fuzzy u (IT2Fu) charts as a new approach to detecting defects. In addition, this paper presents a comparative analysis of performances between traditional u-control charts, type-1 fuzzy u-control charts, and type-2 fuzzy u-control charts. 23 samples of lubricant data with 48 subgroups were examined to identify the defects. The output showed that all of the control charts produced almost similar results except for data 14, which is “out of control” in IT2Fu-control charts but “in control” in traditional u-control charts and “rather in control” in type-1 fuzzy u-control charts. Furthermore, the performances of the charts were compared using a probability-based average run length (ARL), where probability type 1 error is computed. It was found that the ARL value of the IT2Fu-control chart showed the lowest value among the three types of charts. The analysis indicated that the IT2Fu-control chart outperformed the traditional u-control chart and the type-1 fuzzy u-control chart. The results obtained seem to support the idea that IT2Fu-control charts are more sensitive compared to type 1 fuzzy u-control charts and traditional u-control charts, so that IT2Fu-control charts are able to adequately support incomplete and vague data on process control.

Application of Trapezoidal Fuzzy Numbers in the Inventory Problem of Decision Science

The objectives of this research study focus on managing production networks efficiently through a systematic and integrated approach. The central idea is to combine inventory control and transportation scheduling to optimize the entire supply chain. The study introduces the concept of the overtime fuzzy inventory-transportation problem (FITP), specifically aiming to determine an optimal distribution plan that considers overtime to minimize total distribution costs. The use of trapezoidal fuzzy numbers (TrFN) is highlighted as a key methodology for addressing uncertainty and incomplete information in real-world problems. TrFNs are considered a generalization of crisp and fuzzy numbers, providing a flexible framework for modelling imprecision. The process of de-fuzzification is mentioned as a means to convert fuzzy numbers into crisp numbers for practical applications, thereby obtaining clear and precise solutions.

A decision-making system based on case-based reasoning for predicting stroke rehabilitation demands in heterogeneous information environment

Recently, the increasing number of stroke survivors has created a greater demand for rehabilitation services. How to predict rehabilitation demands is important for policy makers in various countries. The demand prediction is essentially a multi-attribute decision-making problem. The diverse forms of information or data are often neglected or simplified in previous studies of decision-making problems. In this paper, heterogeneous multi-attributes including crisp numbers, interval numbers, and linguistic variables are defined to represent decision-relevant information concerning the patient's condition in cases when applied to real-life scenarios. Moreover, a deviation minimization optimization model combining CRITIC method and H2TLWMSM operator is proposed to assign the attribute weights with dual information. The WHMACBR system can be used to predict stroke rehabilitation demands for making the diagnosis and treatment decisions. The proposed decision-making system based on the CBR approach simulates the operations of human memory and reasoning, thus assisting physicians in addressing the ever-increasing demand for rehabilitation services of stroke survivors. Finally, the effectiveness and applicability of the case retrieval process of the presented CBR model are demonstrated through a stroke rehabilitation case study (top 3 cases ranked by similarity to the target case are C6, C2, and C3) and comparison analysis with relevant existing CBR models. This indicates that the proposed decision-making system, by considering heterogeneous multi-attributes and their weights, is capable of providing more comprehensive and rational decision-related information in the field of rehabilitation medicine.

A new decision analysis framework for multi-attribute decision-making under interval uncertainty

In decision-making analysis, the description on decision makers' risk attitude under uncertainty is a focus topic. This paper presents a novel decision analysis framework that considers uncertainty and risk using the cumulative prospect theory. The proposed approach describes uncertain preference information using interval-valued fuzzy sets, maintaining this representation without converting intervals into crisp numbers throughout the entire process. To quantify the distance between interval-valued fuzzy sets, we introduce a comprehensive interval-valued distance model that accounts for multiple situations determined based on the relative positions between intervals. Afterwards, some related theorems about the proposed interval-valued distance model are explored mathematically. Then, with the aid of different reference points and interval-valued distance model, the overall interval-valued prospect value of each alternative is investigated. Finally, an Enhanced Minimax Regret-based (EMR) approach is developed to compare and rank the obtained overall interval-valued prospect values. An illustrative example, accompanied by detailed discussions, is provided to demonstrate the flexibility and superiority of the proposed decision analysis framework.

A New Approach to Solve Transportation Problems Under Neutrosophic Environment

The transportation problem has received a lot of attention in the field of operations research. In many circumstances, transportation planners may lack clear information on supply, demand, and transportation costs. Fuzzy sets can accept incomplete information by allowing for degrees of membership, which describe the level of certainty or uncertainty associated with each parameter. Three components—truth-membership, indeterminacy-membership, and falsity-membership degrees—are added to fuzzy numbers to create neutrosophic fuzzy numbers, which enables a more complex depiction of uncertainty. In this paper, we discuss the fuzzy transportation problem in a neutrosophic environment. Here the transportation costs, demands, and supplies are represented by neutrosophic trapezoidal fuzzy numbers. The neutrosophic trapezoidal fuzzy numbers are transformed into crisp numbers by using a ranking function and providing numerical examples to show the proposed method's efficiency to get the minimum optimal cost. Finally, we have demonstrated that our proposed method produced a better optimal solution to existing approaches by comparing its results to those of the existing ones.

A novel approach for solving the triangular fuzzy neutrosophic assignment problem

The assignment problem (AP) is a fundamental challenge in linear programming and operations research focused on optimizing assignments to minimize costs or maximize profits. This study extends the traditional AP to address uncertainties and ambiguities through the neutrosophic assignment problem (NAP). Further advancing this concept, the triangular fuzzy neutrosophic assignment problem (TFNAP) is introduced, utilizing triangular fuzzy neutrosophic numbers (TFNNs) to represent fuzziness and neutrosophy, thus offering a more comprehensive depiction of uncertainty and indeterminacy. The primary aim of this research is to identify the optimal assignment that either minimizes cost or maximizes profit within the TFNAP framework, a task known for its computational complexity. The TFNAP is represented through a triangular fuzzy neutrosophic assignment matrix (TFNAM), which employs TFNNs as its core elements. This study develops and refines algorithms tailored to address these complexities, ensuring originality through a stepwise procedure that simplifies computations in a neutrosophic manner. The methodology includes resolving the problem using a neutrosophic approach and incorporating a score function to convert triangular fuzzy neutrosophic values to their equivalent crisp numbers for comparative purposes. The proposed method?s effectiveness is validated through its application to real-world problems, with the results compared against previously established solutions. The findings demonstrate that the TFNAP framework provides more accurate and insightful outcomes in dealing with uncertainties compared to traditional methods. This study introduces significant innovations in handling ambiguity and indeterminacy in assignment problems, offering a robust tool for optimization in complex, uncertain environments. Conclusively, the developed approach not only enhances the understanding of neutrosophic assignments but also presents a practical solution for real-world applications.

Book depreciation methods under intuitionistic fuzzy environment

In general, depreciation is the monetary value of wear and tear on capital goods inherited from previous years while creating goods and services as a result of production activities. The subject of depreciation is the addition of the loss of value that will occur in the economic values that are thought to be used in a company for more than one year. The most common depreciations methods, straight-line method, declining balance method, and unit-of-production method, are almost always handled with crisp numbers in the literature. However, the uncertainty in the parameters such as initial invest cost, salvage value, and useful life, requires the usage of fuzzy sets to represent the vagueness in these values more realistically. For this purpose, in this study, the depreciation methods are developed under intuitionistic fuzzy environment. The main contribution of this paper to the literature is that it presents the depreciation methods under intuitionistic environment with new equations for the first time. The proposed models are applied to an autonomous trolley problem to illustrate the given approach.

A hybrid generalized TODIM approach for sustainable 3PRLP selection in electronic manufacturing industry

Under the pressure of globalization, environmental protection/resource consumption, and the intensification of market competition, various industries have recognized the significant importance of sustainable supply chain management (SCM), in which reverse logistics (RL) has attracted many concentrations from governments to enterprises and the public. However, it is relatively difficult to operate RL autonomously for most electronic manufacturing companies whose core businesses and competition advantages are not in logistics operations because of limited resources and professional skills. Therefore, it is a good choice to escape from the predicament by selecting an appropriate third-party reverse logistics provider (3PRLP) with professional skills for outsourcing and accomplishing the RL business. Generally, evaluation and selection of a 3PRLP is regarded as a multi-attribute group decision-making (MAGDM) process, in which four aspects, namely, the evaluation attributes system, weight of experts and attributes, preference information representation and alternative ranking method have impacts on the decision-making results. Therefore, this paper constructs an MAGDM framework named generalized TODIM (TOmada de DecisãoInterativa e Multicritério, Portuguese acronym for Interactive multicriteria decision-making) under a hybrid information environment, which includes crisp numbers, interval numbers and fuzzy linguistic terms, for solving the problem. It provides a useful reference for manufacturing enterprises to find out the optimal 3PRLP from many alternatives. In particular, the use of different information forms for different evaluation attributes can not only facilitate decision makers to provide their judgments, but also effectively retain the original decision information as much as possible. Utilization of the generalized TODIM method is helpful for considering the risk attitudes and psychological preference behaviors of decision-makers. Finally, to demonstrate the feasibility and effectiveness of the proposed method, a case study relates to select a sustainable 3PRLP in an electronic manufacturing company was conducted. The findings show that the company should pay more attention to the attributes related to social responsibility, service quality and cost when selecting alternative 3PRLP under the sustainable SCM environment. It also provides useful suggestions for 3PRLP themselves to improve their competitiveness from these aspects. Moreover, the sensitivity and comparative analysis demonstrated the necessity and superiorities of comprehensively considering weights of experts and attributes, adopting hybrid information formats, and taking into account the risk attitudes and psychological preference behaviors of decision-makers during the evaluation and selection process.

Fuzzy Dijkstra Algorithm to Find the Shortest Path

Shortest path for crisp numbers can be found efficiently using the Dijkstra’s algorithm. There are some uncertain conditions in the environment which introduces a fuzzy number for the distances between two nodes. A classical Dijkstra algorithm is modified to fuzzy Dijkstra algorithm to address the problem. Graded mean integration is used to enhance the algorithm by calculating and comparing the path length between two nodes when the arc length is fuzzy. A sample network is shown to understand the shortest path for fuzzy numbers. Key Words: dijkstra’s algorithm, fuzzy numbers, graded mean integration

Risk assessment based on failure mode and effects analysis (FMEA) and WASPAS methods under probabilistic double hierarchy linguistic term sets

Failure mode and effects analysis (FMEA) is regarded as an efficient tool in identifying and eliminating system failures and important tool in safety and reliability analysis. However, the traditional FMEA method has some drawbacks, for example, in the process of risk assessment, traditional FMEA method usually ignores the influence factors of weight. Simultaneously, when represent risk assessment information, crisp numbers and fuzzy sets cannot represent complex linguistic information forms accurately. Moreover, the traditional FMEA method ranks risk failures through simple product. Thus, based on these disadvantages, this paper aims to present an improved FMEA method combines with the Weighted Aggregates Sum Product Assessment (WASPAS) method and weight determination method to address the risk assessment problems under probabilistic double hierarchy linguistic (PDHL) environment. In assessment process, the probabilistic double hierarchy linguistic term sets (PDHLTSs) can indicate the probability distribution as a linguistic tool to describe the information reasonably and precisely. The contribution s of this paper can be incarnated as below. Firstly, the PDHLTSs are utilized to provide the linguistic information of failure modes and construct risk evaluation model in risk assessment. Secondly, FMEA method is combined with WASPAS method to overcoming some drawbacks existing in traditional FMEA such as without consideration of the weight of three aspects in Severity (S), Occurrence (O) and Detection (D) and too simple to evaluate the risk factors using Risk Priority Number (RPN) values. Meanwhile, in response to the issue of traditional FMEA method neglecting expert weights, weight determination method is established based on the SNA method under the PDHL context. Additionally, the FMs are ranked through the multi-attribute decision-making method WASPAS, and the final risk ranking result is obtained. Finally, the proposed method is applied to the marine risk assessment faced in the development of marine economy. Based on the assessment results, suggestions are given for the risk problems. In addition, we also carried out sensitivity analysis and accuracy analysis on the evaluation results. At the same time, comparison analysis with existing methods is given to illustrate the superiority of the proposed method.