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Solving Intuitionistic Fuzzy Solid Transportation Problem Via New Ranking Method Based on Signed Distance
Abstract
In this paper, signed distance of Symmetrical Intuitionistic Fuzzy Numbers (SIFNs) is introduced. Based on this signed distance and the crisp ranking system on real numbers, a new ranking system for SIFNs is defined, which seems to be very realistic. To illustrate the applicability and suitability of the proposed ranking method and to deal with ambiguity and imprecision, one of the vital mathematical programming problem viz. Solid Transportation Problem (STP) is formulated in intuitionistic fuzzy environment. A new method has been proposed to compute initial basic feasible solution for the same. Also the significance of the proposed approach over existing methods is illustrated. Finally numerical examples are solved to demonstrate the efficiency of the proposed methods.
... Kundu et al. [15] studied multi-objective STP under different uncertain environment, in which the unit transportation costs are represented as fuzzy, random, and hybrid variables, respectively. Numerous researchers presented their work on STP by introducing new method, for example, Sinha et al. [16], Aggarwal and Gupta [17], Sinha et al. [16], etc. addressed a novel concept regarding the TP where they maximized the profit and minimized the transporting time subject to constraints. They considered all the parameters as trapezoidal interval type-2 fuzzy numbers. ...
... They considered all the parameters as trapezoidal interval type-2 fuzzy numbers. Aggarwal and Gupta [17] introduced a new ranking system for signed distance of intuitionistic fuzzy numbers and formulated an STP in intuitionistic environment to compute initial basic feasible solution. Acharya et al. [18] applied an interactive fuzzy goal programming approach for solving multi-objective generalized STP. ...
This article attempts to study cost minimizing multi-objective fractional solid transportation problem with fuzzy cost coefficients, fuzzy supply quantities, fuzzy demands, and/or fuzzy conveyances. The fuzzy efficient concept is introduced in which the crisp efficient solution is extended. A necessary and sufficient condition for the solution is established. Fuzzy geometric programming approach is applied to solve the crisp problem by defining membership function so as to obtain the optimal compromise solution of a multi-objective two-stage problem. A linear membership function for the objective function is defined. The stability set of the first kind is defined and determined. A numerical example is given for illustration and to check the validity of the proposed approach.
... Kundu et al. [15] studied multi-objective STP under different uncertain environment, in which the unit transportation costs are represented as fuzzy, random, and hybrid variables, respectively. Numerous researchers presented their work on STP by introducing new method, for example, Sinha et al. [16], Aggarwal and Gupta [17], Sinha et al. [16], etc. addressed a novel concept regarding the TP where they maximized the profit and minimized the transporting time subject to constraints. They considered all the parameters as trapezoidal interval type-2 fuzzy numbers. ...
... They considered all the parameters as trapezoidal interval type-2 fuzzy numbers. Aggarwal and Gupta [17] introduced a new ranking system for signed distance of intuitionistic fuzzy numbers and formulated an STP in intuitionistic environment to compute initial basic feasible solution. Acharya et al. [18] applied an interactive fuzzy goal programming approach for solving multi-objective generalized STP. ...
This article attempts to study cost minimizing multi-objective fractional solid transportation problem with fuzzy cost coefficients, fuzzy supply quantities, fuzzy demands, and/or fuzzy conveyances. The fuzzy efficient concept is introduced in which the crisp efficient solution is extended. A necessary and sufficient condition for the solution is established. Fuzzy geometric programming approach is applied to solve the crisp problem by defining membership function so as to obtain the optimal compromise solution of a multi-objective two-stage problem. A linear membership function for the objective function is defined. The stability set of the first kind is defined and determined. A numerical example is given for illustration and to check the validity of the proposed approach.
... Later, in 1989, Atanassov and Gargov have also introduced the interval-valued intuitionistic fuzzy set by using interval valued membership and nonmembership functions (Atanassov and Gargov 1989). In 2016, ranking method has also been used to deal intuitionistic fuzzy solid transportation problems (Aggarwal and Gupta 2016). Sometimes, the problem evolves when the data are collected from partial or absolute contradictory information and knowledge. ...
Cubic n-Inner Product Space and Intuitionistic cubic fuzzy linear space
... Later, in 1989, Atanassov and Gargov have also introduced the interval-valued intuitionistic fuzzy set by using interval valued membership and nonmembership functions (Atanassov and Gargov 1989). In 2016, ranking method has also been used to deal intuitionistic fuzzy solid transportation problems (Aggarwal and Gupta 2016). Sometimes, the problem evolves when the data are collected from partial or absolute contradictory information and knowledge. ...
... Later, in 1989, Atanassov and Gargov have also introduced the interval-valued intuitionistic fuzzy set by using interval valued membership and nonmembership functions (Atanassov and Gargov 1989). In 2016, ranking method has also been used to deal intuitionistic fuzzy solid transportation problems (Aggarwal and Gupta 2016). Sometimes, the problem evolves when the data are collected from partial or absolute contradictory information and knowledge. ...
It proposes the PSK (P. Senthil Kumar) method for solving intuitionistic fuzzy solid transportation problems (IFSTPs). In our daily life, uncertainty comes in many ways, e.g., the transportation cost (TC) is not a fixed one, it varies from time to time due to market conditions (i.e., the price of diesel is depending on the cost of crude oil), mode of the transportation, etc. So, to deal with the TP having uncertainty and hesitation in TC, in this chapter, the author divided IFSTP into 4 categories and solved type II- IFSTP by using TIFNs. The model of type II- IFSTP and its relevant CSTP both are presented. The PSK method is presented clearly with the proof of some theorems and corollary. To illustrate the PSK method with proposed models, the numerical experiment and its related graphs are presented. Real-life problems are identified and solved by the PSK method with MATLAB and LINGO software. Analysis, discussion, merits, and demerits of the PSK method are all presented. A valid conclusion and recommendations are given. Finally, some of the future research areas are also suggested.
Purpose
This research attempts to present a solid transportation problem (STP) mechanism in uncertain and indeterminate contexts, allowing decision makers to select their acceptance, indeterminacy and untruth levels.
Design/methodology/approach
Due to the lack of reliable information, changeable economic circumstances, uncontrolled factors and especially variable conditions of available resources to adapt to the real situations, the authors are faced with a kind of uncertainty and indeterminacy in constraints and the nature of the parameters of STP. Therefore, an approach based on neutrosophic logic is offered to make it more applicable to real-world circumstances. In this study, the triangular neutrosophic numbers (TNNs) have been utilized to represent demand, transportation capacity, accessibility and cost. Then, the neutrosophic STP was converted into an interval programming problem with the help of the variation degree concept. Then, two simple linear programming models were extracted to obtain the lower and upper bounds of the optimal solution.
Findings
The results reveal that the new model is not complicated but more flexible and more relevant to real-world issues. In addition, it is evident that the suggested algorithm is effective and allows decision makers to specify their acceptance, indeterminacy and falsehood thresholds.
Originality/value
Under the transportation literature, there are several solutions for TP and STP in crisp, fuzzy set (FS) and intuitionistic fuzzy set (IFS) conditions. However, the STP has never been explored in connection with neutrosophic sets to the best of the authors’ knowledge. So, this work tries to fill this gap by coming up with a new way to solve this model using NSs.
Multi-objective optimization is an emerging field concerning optimization problems associated with more than one objective function, each of them has to be optimized simultaneously. Multi-objective optimization is widely used in logistics and supply chains to reduce the cost and time involved in transportation. With the increase in Global Supply Chains, many organizations are facing the challenges of delivering products to their customers at a fast pace, low cost, and high reliability. There are numerous factors that may affect the goal of an organization to optimize the cost, time, and effort during the transportation of their products to the end customers. For instance, in the existing transportation problems, the type of vehicles used for the movement of the products is not focused. Transportation of the goods is considered to utilize any type of vehicle irrespective of the nature of the goods. However, in real-life scenarios, there are certain constraints in the vehicle used to transport the finished goods or raw materials from a source to a destination. Vehicles such as tanker trucks, top open trucks, closed trucks, etc. need to be booked based on the nature of goods to be transported. Also, the cost and time of transportation are uncertain in nature. In this paper, we formulate the Multi-Objective Solid Transportation Problem (MOSTP) by considering the above issue. The uncertain parameters of the problem are considered as Pentagonal Intuitionistic Fuzzy Numbers (PIFN). Magnitude method is used for defuzzification. An algorithm to find the solution of formulated Intuitionistic Fuzzy Multi-Objective Solid Transportation problem (IFMOSTP) is provided. The proposed model is illustrated by a numerical example which is solved with the help of LINGO software.
This paper investigates a periodic review fuzzy inventory model with lead time, reorder point, and cycle length as decision variables. The main goal of this study is to minimize the expected total annual cost by simultaneously optimizing cycle length, reorder point, and lead time for the whole system based on fuzzy demand. Two models are considered in this paper: one with normal demand distribution and another with a distribution-free approach. The model assumes a logarithmic investment function for lost-sale rate reduction. Furthermore, two separate efficient computational algorithms are explained to obtain the optimal solution. Some numerical examples are given to illustrate the model.
In this paper we define division operation of Triangular Intuitionistic Fuzzy number (TIFN) using α,β – cut and a scoring function to rank TIFNs. An accuracy function to defuzzify TIFN is also introduced. Based on this new approach, the solution of Intuitionistic Fuzzy Linear Programming is obtained. Finally, an illustrative example is given to verify the developed approach.
Techniques for ranking simple fuzzy numbers are abundant in nature. However, we lack effective methods for ranking intuitionistic fuzzy numbers(IFN). The aim of this paper is to introduce a new ranking procedure for trapezoidal intuitionistic fuzzy number(TRIFN). To serve the purpose, the value and ambiguity index of TRIFNs have been defined. In order to rank TRIFNs, we have defined a ranking function by taking sum of value and ambiguity index. To illustrate the the proposed ranking method a numerical example has been given.
A definition of the concept ‘intuitionistic fuzzy set’ (IFS) is given, the latter being a generalization of the concept ‘fuzzy set’ and an example is described. Various properties are proved, which are connected to the operations and relations over sets, and with modal and topological operators, defined over the set of IFS's.
1 In this paper I will introduce the trapezoidal intuitionistic fuzzy numbers (IF numbers) and will prove some operations for them. Also I am going to propose a new ordering method for IF numbers in which I will consider two characteristic values of membership and non-membership functions for an IF number. These values are defined by the integral of the inverse fuzzy membership and non-membership functions multiplied by the grade with powered parameter. I will introduce the definition of intuitionistic fuzzy numbers (IFN) and make some operations on IF numbers. In section 4, I will review some ranking methods for intuitionistic fuzzy numbers. Then the characteristic value for fuzzy number introduced by Chiao [3] is generalized in the case of intuitionistic fuzzy numbers and I will propose a ranking method for IF numbers in section 5. 2. Basic definitions and properties In this section I will review the basic concepts of fuzzy numbers and intuitionistic fuzzy sets.
There are several methods in the literature to deal with the sensitivity analysis of fuzzy linear programming (FLP) problems. In this paper, the shortcomings of some existing methods are pointed out and to overcome these shortcomings, a new method is proposed to handle the sensitivity analysis of that kind of FLP problems in which the elements of the coefficient matrix of the constraints are represented by real numbers, and the remaining parameters are represented by symmetric trapezoidal fuzzy numbers. To the best of our knowledge, till now there is no method in the literature to deal with the sensitivity analysis of such FLP problems without converting them to the crisp linear programming problems. The advantages of the proposed method over existing methods are discussed. To illustrate the approach a numerical example is examined by using the proposed method and the obtained results are discussed. The method is easy to understand and to apply for dealing with the sensitivity analysis of the same problems occurring in real life situations.
This paper illustrates the relationship between quality improvement, reorder point, and lead time, as affected by backorder rate, in an imperfect production process. To reduce the total system cost by optimizing the setup cost, lot size, lead time, reorder point, and process quality parameter simultaneously, we first consider that the lead time demand follows a normal distribution, then we apply the distribution free approach for the lead time demand. We prove two lemmas which are used to find optimal solutions for the basic and distribution free models. We compare our models with the existing models using numerical examples and show that significant savings over the existing models can be achieved.