Algorithms and Software Packages for Solving Transportation Problems With Intuitionistic Fuzzy Numbers

In this chapter, two different methodologies are proposed to find out the optimal solution to the balanced and unbalanced intuitionistic fuzzy transportation problems (UBIFTPs). In addition, the parameter of both the balanced and UBIFTPs are considered to be triangular intuitionistic fuzzy numbers (TIFNs). Two new methodologies, respectively method-1 and method-2, are presented in this chapter. Proposed method-1 is based on linear programming technique, and proposed method-2 is based on modified distribution method. Both the methodologies are used to solve the balanced and UBIFTPs. The ideas of the proposed methodologies are illustrated with the help of real-life numerical examples. The solutions obtained by the proposed methodologies are checked with some software (e.g., MATLAB, LINGO) and the computer code related to the proposed problems is also given. The unique results, comparative study, discussions, and the merits of the proposed methodologies are all given. At the end of the chapter, future work is mentioned.

In dealing with real world transportation problems, several issues are frequently encountered as a result of uncontrollable factors. To deal this uncertainties, many authors have suggested transportation problems with intuitionistic fuzzy parameters. In this article, fully intuitionistic fuzzy transportation problem (InFTP) is considered in which the parameters are triangular intuitionistic fuzzy numbers. To solve this, initially intuitionistic fuzzy branch and bound technique are applied to get the initial basic feasible solution and then intuitionistic fuzzy modified distribution (InFMODI) method is applied to acquire the optimal solution of the fully InFTP. Also, a new ordering is developed here in which some properties of compensation, linearity, additive and partial order relations are satisfied. The optimal solution obtained from the proposed method satisfies the condition of optimality and feasibility. Finally, two numerical examples are provided to show the effectiveness of the proposed method.

In today’s real world problems such as in corporate or in industrial, the experts and the decision makers have to suffer with uncertainty as well as with hesitation usually, due to the complexity of the situations. The main reasons behind these complexities are lack of good communications with all involved persons, error in data’s, understanding of markets, unawareness of customers etc. So, In this paper, we consider a transportation problem having uncertainty and hesitation in supply and demand. We formulate the problem and utilize triangular intuitionistic fuzzy numbers (TIFNs) to deal with uncertainty and hesitation. We propose intuitionistic fuzzy methods to find starting basic feasible solution in terms of TIFNs. Intuitionistic fuzzy modified distribution method has been proposed to find optimal solution. The shortcomings of the existing methods have been pointed out. The proposed method is illustrated by numerical examples.

A fuzzy transportation problem (FTP) includes cost, supply and demand of transportation problems. Its numbers are fuzzy numbers. Fuzzy transportation problem works to reduce transportation cost of some commodities through a capacitate network. Present research paper points out a technique with an alpha cut, optimal solution for solving transportation problem. We suggest a technique to find the fuzzy optimal solution on scales of transportation problem and propose a new hexagonal representation of fuzzy numbers. In general, the comparison of balanced fuzzy transportation problem (BFTP) and unbalanced fuzzy transportation problem (UFTP) shows that the optimal transportation cost of UFTP is less than BFTP.

The urge to remanufacture and address environmental concerns in various industrial processes has drawn the attention of academics as well as practitioners towards Closed-loop Supply Chain Networks (CLSC). Although everchanging and complex external factors including social and economic ones, adversely impact the sustainable development of closed-loop supply chain networks. The basic aim of the research is to optimize the functioning of CLSC networks. For the above-said, two objective functions are made. The first objective is to minimize the cost of production and assembly expenses of the forward and reverse logistics. Secondly, an endeavour has been made to reduce the fixed costs associated with plants and retailers. For the sake of achieving two objective functions, two methods are employed: triangular fuzzy numbers and triangular intuitionistic fuzzy numbers. Among the two methods, triangular intuitionistic fuzzy numbers achieved the said objectives with greater optimization substantiated by statistics. This method can deal with uncertain external factors without undermining the optimization of the CLSC networks.

In this paper a new method is proposed for finding an optimal solution for Intuitionistic Fuzzy Transportation Problem, directly. Here the supply and demand are triangular intuitionistic fuzzy numbers. A new ranking method is introduced. The solution procedure is illustrated with numerical example.

A portfolio selection model with return as triangular intuitionistic fuzzy number is developed in this study to assess and select portfolios on China Stock Exchange. Although the portfolio selection has been widely investigated, and most studies have regarded return and risk as the main decisive criteria, there are many uncertainties in the financial market, such as social, political and human psychological factors, which makes it difficult for us to describe the risks in line with empirical evidence. To fill this gap, first, a literature review was conducted to clarify the current situation and shortcomings of portfolio selection. Second, the triangular intuitionistic fuzzy number was used to fuzzify the coefficients of the objective function and the constraints in the portfolio model. Third, the triangular intuitionistic fuzzy number model was transformed into a linear programming model by using the selected exact ranking function, and the model was solved by MATLAB. Finally, the historical monthly returns of 10 stocks in China’s Stock Exchange from December 2017 to November 2020, lasting 36 months, were selected to demonstrate the model. The results indicate that the portfolio selection model with triangular intuitionistic fuzzy number can better meet the uncertainty of the real securities market, and more reasonable investment decision guidance can be provided for investors. Besides, the limitations of this study are pointed out, and implications and directions for future research are discussed.

This paper is concerned with the solution procedure of a Transportation Problem in which costs are triangular intuitionistic fuzzy numbers (TIFN) and availabilities and demands are taken as...

Business environment is characterized by greater domestic and international competitive position in the global market. Vendors play a key role in achieving the so-called corporate competition. It is not easy however to identify good vendors because evaluation is based on multiple criteria. In practice, for VSP most of the input information about the criteria is not known precisely. Intuitionistic fuzzy set is an extension of the classical fuzzy set theory (FST), which is a suitable way to deal with impreciseness. In other words, the application of intuitionistic fuzzy sets instead of fuzzy sets means the introduction of another degree of freedom called nonmembership function into the set description. In this paper, we proposed a triangular intuitionistic fuzzy number based approach for the vendor selection problem using analytical hierarchy process. The crisp data of the vendors is represented in the form of triangular intuitionistic fuzzy numbers. By applying AHP which involves decomposition, pairwise comparison, and deriving priorities for the various levels of the hierarchy, an overall crisp priority is obtained for ranking the best vendor. A numerical example illustrates our method. Lastly a sensitivity analysis is performed to find the most critical criterion on the basis of which vendor is selected.

A two-compartment drug concentration model using intuitionistic fuzzy sets

In the present paper, a novel intuitionistic fuzzy Multiple Attribute Decision Making (MADM) is proposed for modelling and solving analytical hierarchy process (AHP) problems with small amount of relationship among various criteria. Assigning a membership degree, fuzzy sets can model some uncertainty to the decision space. Intuitionistic fuzzy sets model the uncertainty more accurately associated with both membership and non-membership degree. Based on advantages of Intuitionistic fuzzy sets, this paper first uses IF-AHP to evaluate the weighting for each criterion and then develops an intuitionistic fuzzy DEMATEL method to establish contextual relationships among those criteria. Finally, an integrated IF-DEMATEL-AHP method is proposed and used for a case study for selecting managers in the automobile industry in Iran.

Supplier selection is a strategic complex multi-criteria problem of selecting the appropriate supplier. The main objective of this paper is to propose a hybrid model including intuitionistic fuzzy AHP using triangular intuitionistic fuzzy numbers and goal programming for the supplier selection problem. Intuitionistic fuzzy AHP is a useful tool for handling the complex problems with hierarchical form under intuitionistic fuzzy environment. And the goal programming is a linear programming based method and deals with multiple conflicting objective measures effectively. In this study, we evaluated three suppliers according to five criteria by using intuitionistic fuzzy AHP which utilizes triangular intuitionistic fuzzy scale. By applying intuitionistic fuzzy AHP, the priority weights of the criteria and suppliers are calculated. Then, the weights are used in the goal programming model to find the best alternatives. The results point out that the supplier selection problem can be tackled with using the proposed hybrid method effectively.

In scenarios where system component failure probabilities are insufficient or imprecise, classical fault tree analysis methods often fail in providing precise assessments of system reliability and safety. This limitation arises because classical fault tree analysis relies on precise data and probabilistic measures, which may not adequately capture inherent uncertainties and vagueness in real-world environments. Therefore, a general approach is required to reliably predict and diagnose system failures that incorporates the uncertainties inherent in failure data. The fuzzy sets effectively manage imprecise information and serve as a robust foundation to address ambiguity in real-world failure scenarios. Additionally, the use of intuitionistic fuzzy sets further enhances this framework by considering both membership and non-membership degrees, thereby providing a more comprehensive and precise reliability assessment. Therefore, in the current study, the intuitionistic fuzzy set theory is proposed to handle the inherent uncertainty in reliability assessments. Furthermore, triangular intuitionistic fuzzy numbers are used to represent the failure probabilities of the system components. The proposed method integrates intuitionistic fuzzy set theory with fault tree analysis to determine the reliability and failure probability of a rope on a crane, and the computed results are compared with the results of existing methods.

The fuzzy Analytical Hierarchy Process (AHP) is a method in fuzzy decision-making, which helps decision makers make the best choice of alternatives when dealing with uncertainty. The objective of this paper is to propose a conversion method of trapezoidal intuitionistic fuzzy numbers (TrIFN) into interval-valued intuitionistic fuzzy numbers (IVIFN), followed by a new fuzzy AHP based on the proposed approximation. The linguistic judgments of decision makers are converted into TrIFN before being transformed into IVIFN using the nearest weighted intuitionistic interval approximation (NWIIA). The proposed model is implemented in supplier selection problems by taking into account five criteria comprising cost, quality, service, cycle time, and reputation. The results showed that the proposed model simplifies the fuzzy AHP using TrIFN without affecting the consistency of the solutions from the existing method, thus supporting the fact that TrIFN has a better capability of processing ill-defined quantities. This is because TrIFN entails a generalization of the triangular intuitionistic fuzzy number (TIFN), making it a better form of representation of linguistic variables for reducing the inexactness.

The aim of this paper is to develop a new methodology for solving bi-matrix games with payoffs of Atanassov’s intuitionistic fuzzy (IF) numbers (IFNs), which are called IF bi-matrix games for short. In this methodology, we propose a weighted mean-area ranking method of IFNs, which is proven to satisfy the linearity. Hereby, the concept of Pareto optimal solution of IF bi-matrix games is introduced and the Pareto optimal solution can be obtained through solving the parameterized non-linear programming model, which is derived from an IF mathematical programming model based on the proposed weighted mean-area ranking method of IFNs. Validity and applicability of the model and method proposed in this paper are illustrated with a practical example of two commerce retailers’ strategy choice problem.