It presents the type-2 intuitionistic fuzzy solid transportation problem (type-2 IFSTP) in a real-world situation and introduces an efficient technique based on software codes for solving type-2 IFSTP. It defines various types of intuitionistic fuzzy solid transportation problems (IFSTPs) and presents a mathematical model of full and type-2 IFSTPs. Additionally, the theorem, corollary, and their proof, as well as some results relevant to type-2 IFSTPs, are presented. Both the existing problem’s solution and the proposed type-2 IFSTP’s solution are compared using software (e.g., Matlab). In particular, solving IFSTPs using Matlab is new in the literature. Shortcomings in the existing results are pointed out and solved. Using the proposed approach, the optimal solution and optimal objective value of both the solid transportation problem (STP) and type-2 IFSTPs are obtained concurrently with one LP model, which is the key objective of this article. It also discusses the time complexity of the existing algorithms for solving STPs with different intuitionistic fuzzy numbers (IFNs). Both the proposed results and the benefits of the proposed methodology are thoroughly discussed. Finally, a sound conclusion and future research directions are provided.

We have several logics to obtain the fuzzy optimal solution for fuzzy transportation problems (FTPs). But there is no logic to solve transportation problem (TP) under four different uncertain environments using a particular method in the literature. So, the author divided the TP under four different environments and formulates the problems and utilizes the fuzzy numbers to solve them. A method, namely PSK (P. Senthil Kumar) method, to find the fuzzy optimal solutions to the FTPs is proposed. Applications of the PSK method compared to other existing methods are demonstrated with four different numerical examples. To illustrate the PSK method, different types of FTPs are solved by using the PSK method with the help of software, for example, Lingo, Matlab, Tora, RGui, and RStudio, and the obtained results are compared, analyzed, and discussed. Conclusions on the study and recommendations for future research are also given.

This article addresses the three-dimensional mixed intuitionistic fuzzy assignment problems (3D-MIFAPs). In this article, firstly, the author formulates an assignment problem (AP) and assumes the parameters are in uncertainty with hesitation. Secondly, based on the nature of the parameter, the author defines various types of solid assignment problem (SAP) in uncertain environments. Thirdly, to solve 3D-MIFAP, the PSK method for finding an optimal solution of fully intuitionistic fuzzy assignment problem (FIFAP) is extended by the author. Fourthly, the author presents the proofs of the proposed theorems and corollary. Fifthly, the proposed approach is illustrated with three numerical examples, and the optimal objective value of 3D-MIFAP is obtained in the form of intuitionistic fuzzy number, and the solution is checked with MATLAB, and their coding are also given by the author. Sixthly, the author presents the comparison results and their graphical representation, merits and demerits of the proposed and existing methods, and finally, a conclusion and future research directions.

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.

The assignment problem (AP) is a particular case of a linear programming problem that deals with the allocation of various resources for various activities on a 1-to-1 basis. It does so in such a manner that the profit or sale involved in the process is maximum and cost or time is minimum. Generally, the profit/sale/cost/time is called the parameter of the AP and this is not a crisp number due to some uncontrollable factors. They can also involve uncertainty and hesitation. Therefore, to solve the AP under an intuitionistic fuzzy environment in this chapter, the author proposes the PSK (P. Senthil Kumar) method. Numerous theorems which are related to intuitionistic fuzzy assignment problem is proposed and is proved by PSK. By using the PSK method, the real-life related fully intuitionistic fuzzy assignment problems are solved. The proposed results are verified by both LINGO 17.0 and TORA software packages. In addition to verifying the efficiency and realism of the proposed method, the computer code based on LINGO 17.0 is presented. Results, discussion, comparative study, and the advantages of the PSK method are given. The chapter ends with the conclusion and future studies.

In conventional transportation problem, supply, demand and costs are fixed crisp numbers. Therefore, in this situation, the decision maker (DM) can predict transportation cost exactly. On the contrary, in real world transportation problems the costs are in uncertain quantities with hesitation due to various factors like variation in rates of fuels, traffic jams, weather in hilly areas, etc. In such situations, the DM cannot predict transportation cost exactly and it will force the DM to hesitate. So, to counter these uncertainties, in this article, the author designed a transportation problem in which supplies, demands are crisp numbers and cost is intuitionistic fuzzy number. This type of problem is termed as type-2 intuitionistic fuzzy transportation problem (type-2 IFTP). Hence to deal with uncertainty and hesitation in transportation problem, intuitionistic fuzzy zero point method is proposed to find out optimal solution to the type-2 IFTP. Moreover, special kind of type-2 IFTP is proposed and its related theorems are proved. Finally, the ideas of the proposed method are illustrated with the help of numerical example which is followed by the results and discussion.

In this article, the crisp, fuzzy and intuitionistic fuzzy optimization problem is formulated. The basic definitions and notations related to optimization problems are given in the preliminaries section. Algorithms for solving the optimization problems using fuzzy and intuitionistic fuzzy set is presented in this article. Then, with the help of the proposed algorithm the optimal solution of the crisp, fuzzy and intuitionistic fuzzy optimization problems are determined. A new theorem related to type-2 fuzzy/type-2 intuitionistic fuzzy optimization problems is proposed and proved. Some new and concrete results related to type-2 fuzzy/type-2 intuitionistic fuzzy optimization problems are presented. To illustrate the proposed method, some real-life numerical examples are presented. The proposed article provides seven fully worked examples with screenshots of output summaries from the software used in the computations for better understanding. The advantages of the proposed approach as compared to other existing work are also specified. Detail analyses of the comparative study as well the discussion are given. To show the advantages of the proposed approach, superiority analysis is discussed. Comparison analysis and the advantages of the proposed operators are also discussed. Some managerial applications and the advantages of the proposed approach are given. Finally, conclusion and future research directions are also given.

When people solve real-life SAP they tend to face the uncertainty state as well as hesitation due to many uncontrollable factors. To deal with uncertainty and hesitation many authors have suggested the intuitionistic fuzzy representation for the data. In this article, the author tried to categorise the SAP under uncertain environment. He formulates the IFSAP and utilizes the TIFN to deal with uncertainty and hesitation. The SAP has uncertainty and hesitation in cost/time/profit/production is known as FIFSAP. The PSK (P. Senthil Kumar) method for finding an optimal solution for FIFAP is extended to solve the FIFSAP and the optimal objective value of FIFSAP is obtained in terms of TIFN. The main advantage of this method is that the optimal solution/assignment of FIFSAP is obtained without using the Hungarian method and intuitionistic fuzzy reduction method. Moreover, the proposed method is computationally very simple and easy to understand. The numerical example is presented to demonstrate computing procedure. The results affirm efficiency of the proposed method.

In this article, the author categorises the solid transportation problem (STP) under uncertain environments. He formulates the mixed and fully intuitionistic fuzzy solid transportation problems (FIFSTPs) and utilizes the triangular intuitionistic fuzzy number (TIFN) to deal with uncertainty and hesitation. The PSK (P. Senthil Kumar) method for finding an intuitionistic fuzzy optimal solution for fully intuitionistic fuzzy transportation problem (FIFTP) is extended to solve the mixed and type-4 IFSTP and the optimal objective value of mixed and type-4 IFSTP is obtained in terms of triangular intuitionistic fuzzy number (TIFN). The main advantage of this method is that the optimal solution of mixed and type-4 IFSTP is obtained without using the basic feasible solution and the method of testing optimality. Moreover, the proposed method is computationally very simple and easy to understand. Finally, the procedure for the proposed method is illustrated with the help of numerical examples which is followed by graphical representation of the finding.

This paper sustains a sound mathematical and computing background. In this paper, the software-based approach for solving intuitionistic fuzzy solid assignment problem (IFSAP) is presented. The IFSAP is formulated and it is solved by using Lingo 17.0 software tool. Theorems related to IFSAP is proved. The IFSAP and its crisp solid assignment problem both are solved at a time and their optimal solution is obtained. In addition, the optimal objective values of both the IFSAP and its crisp solid assignment problem (SAP) are estimated with the help of substituting the optimal solution(s) to their respective decision variables in the objective functions. Some new and important results are proposed. To illustrate the efficiency of the proposed method the numerical example is presented. The reliability of the proposed results are verified by using the numerical example. Strengths and weakness of the paper is mentioned. The novelty of the analysis is given into a coherent, concise, and meaningful manner of analysis. Social issue (real-life problem) is converted into a mathematical model and it is solved by the proposed method. At the end, the advantages of the proposed algorithm is explained.

In this article, two methods are presented, proposed method 1 and proposed method 2. Proposed method 1 is based on linear programming technique and proposed method 2 is based on modified distribution method. Both of the methods are used to solve the balanced and unbalanced intuitionistic fuzzy transportation problems. The ideas of the proposed methods are illustrated with the help of real life numerical examples which is followed by the results and discussion and comparative study is given. The proposed method is computationally very simple when compared to the existing methods, it is shown to be and easier form of evaluation when compared to current methods.

There are several algorithms, in literature, for obtaining the fuzzy optimal solution of fuzzy transportation problems (FTPs). To the best of the author's knowledge, in the history of mathematics, no one has been able to solve transportation problem (TP) under four different uncertain environment using single method in the past years. So, in this chapter, the author tried to categories the TP under four different environments and formulates the problem and utilizes the crisp numbers, triangular fuzzy numbers (TFNs), and trapezoidal fuzzy numbers (TrFNs) to solve the TP. A new method, namely, PSK (P. Senthil Kumar) method for finding a fuzzy optimal solution to fuzzy transportation problem (FTP) is proposed. Practical usefulness of the PSK method over other existing methods is demonstrated with four different numerical examples. To illustrate the PSK method different types of FTP is solved by using the PSK method and the obtained results are discussed.

In real-life decisions, usually we happen to suffer through different states of uncertainties. In order to counter these uncertainties, in this paper, the author formulated a transportation problem in which costs are triangular intuitionistic fuzzy numbers, supplies and demands are crisp numbers. In this paper, a simple method for solving type-2 intuitionistic fuzzy transportation problem (type-2 IFTP) is proposed and optimal solution is obtained without using intuitionistic fuzzy modified distribution method and intuitionistic fuzzy zero point method. So, the proposed method gives the optimal solution directly. The solution procedure is illustrated with the help of three real life numerical examples. Defect of existing results proposed by Singh and Yadav (2016a) is discussed. Validity of Pandian's (2014) method is reviewed. Finally, the comparative study, results and discussion are given.

Distribution companies face great challenges to organize their fleet efficiently. In this work, a study of capacitated vehicle routing problem with simultaneous pickup and delivery between two depots with several nodes spread in between is taken up. The purpose of this study is to develop a heuristic procedure for the above mentioned problem and analyse the results. The problem consists of two depots with 25 nodes spread in between them. A heuristic method called linear sweep algorithm have been developed to solve the problem. The objectives of this work were to minimize the number of vehicles used and maximize the utilization of vehicle capacity. The proposed heuristic was able to achieve both the objectives and hence found to be useful for solving the above problem in different scenarios.

This article describes how in solving real-life solid transportation problems (STPs) we often face the state of uncertainty as well as hesitation due to various uncontrollable factors. To deal with uncertainty and hesitation, many authors have suggested the intuitionistic fuzzy (IF) representation for the data. In this article, the author tried to categorise the STP under uncertain environment. He formulates the intuitionistic fuzzy solid transportation problem (IFSTP) and utilizes the triangular intuitionistic fuzzy number (TIFN) to deal with uncertainty and hesitation. The STP has uncertainty and hesitation in supply, demand, capacity of different modes of transport celled conveyance and when it has crisp cost it is known as IFSTP of type-1. From this concept, the generalized mathematical model for type-1 IFSTP is explained. To find out the optimal solution to type-1 IFSTPs, a single stage method called intuitionistic fuzzy min-zero min-cost method is presented. A real-life numerical example is presented to clarify the idea of the proposed method. Moreover, results and discussions, advantages of the proposed method, and future works are presented. The main advantage of the proposed method is that the optimal solution of type-1 IFSTP is obtained without using the basic feasible solution and the method of testing optimality.

This article proposes a method for solving intuitionistic fuzzy solid transportation problems IFSTPs in which only the transportation costs are represented in terms of intuitionistic fuzzy numbers IFNs. The remaining parameters, namely: supply, demand and conveyance capacity, are all considered into crisp numbers. This type of STP is called a type-2 IFSTP. When solving the real life solid transportation problems STPs those tend to face the uncertainty state as well as hesitation due to many uncontrollable factors. To deal with uncertainty and hesitation many authors have suggested the intuitionistic fuzzy representation for the data. In this article, the author tried to categorise the STPs under the uncertain environment. He formulates the intuitionistic fuzzy STPs and utilizes the triangular intuitionistic fuzzy number TIFN to deal with uncertainty and hesitation. The PSK P.Senthil Kumar method for finding an intuitionistic fuzzy optimal solution for fully intuitionistic fuzzy transportation problem FIFTP is extended to solve the type-2 IFSTP and the optimal objective value of type-2 IFSTP is obtained in terms of TIFN. The main advantage of this method is that the optimal solution of type-2 IFSTP is obtained without using the basic feasible solution and the method of testing optimality. Moreover, the proposed method is computationally very simple and easy to understand. A case study is presented to illustrate the procedure of the proposed method.

In solving real life assignment problem we often face the state of uncertainty as well as hesitation due to various uncontrollable factors. To deal with uncertainty and hesitation many authors have suggested the intuitionistic fuzzy representations for the data. So, in this paper, the authors consider the assignment problem having uncertainty and hesitation in cost/time/profit. They formulate the problem and utilize triangular intuitionistic fuzzy numbers TIFNs to deal with uncertainty and hesitation. The authors propose a new method called PSK P.Senthil Kumar method for finding the intuitionistic fuzzy optimal cost/time/profit for fully intuitionistic fuzzy assignment problem FIFAP. The proposed method gives the optimal object value in terms of TIFN. The main advantage of this method is computationally very simple, easy to understand. Finally the effectiveness of the proposed method is illustrated by means of a numerical example which is followed by graphical representation of the finding.

In conventional transportation problem TP, supplies, demands and costs are always certain. In this paper, the author tried to categories the TP under the mixture of certain and uncertain environment and formulates the problem and utilizes the crisp numbers, triangular fuzzy numbers TFNs and trapezoidal fuzzy numbers TrFNs to solve the TP. The existing ranking procedure of Liou and Wang is used to transform the type-1 and type-3 fuzzy transportation problem FTP into a crisp one so that the conventional method may be applied to solve the TP. The solution procedure differs from TP to type-1 and type-3 FTP in allocation step only. Therefore, the new method called PSK method and new multiplication operation on TrFN is proposed to find the mixed optimal solution in terms of crisp numbers, TFNs and TrFNs. The main advantage of this method is computationally very simple, easy to understand and also the optimum objective value obtained by our method is physically meaningful. The effectiveness of the proposed method is illustrated by means of a numerical example.

In conventional transportation problem (TP), all the parameters are always certain. But, many of the real life situations in industry or organization, the parameters (supply, demand and cost) of the TP are not precise which are imprecise in nature in different factors like the market condition, variations in rates of diesel, traffic jams, weather in hilly areas, capacity of men and machine, long power cut, labourer's over time work, unexpected failures in machine, seasonal changes and many more. To counter these problems, depending on the nature of the parameters, the TP is classified into two categories namely type-2 and type-4 fuzzy transportation problems (FTPs) under uncertain environment and formulates the problem and utilizes the trapezoidal fuzzy number (TrFN) to solve the TP. The existing ranking procedure of Liou and Wang (1992) is used to transform the type-2 and type-4 FTPs into a crisp one so that the conventional method may be applied to solve the TP. Moreover, the solution procedure differs from TP to type-2 and type-4 FTPs in allocation step only. Therefore a simple and efficient method denoted by PSK (P. Senthil Kumar) method is proposed to obtain an optimal solution in terms of TrFNs. From this fuzzy solution, the decision maker (DM) can decide the level of acceptance for the transportation cost or profit. Thus, the major applications of fuzzy set theory are widely used in areas such as inventory control, communication network, aggregate planning, employment scheduling, and personnel assignment and so on.

In solving real life transportation problem we often face the state of uncertainty as well as hesitation due to various uncontrollable factors. To deal with uncertainty and hesitation many authors have suggested the intuitionistic fuzzy representation for the data. So, in this paper, we consider a transportation problem having uncertainty and hesitation in supply, demand and costs. We formulate the problem and utilize triangular intuitionistic fuzzy numbers (TrIFNs) to deal with uncertainty and hesitation. We propose a new method called PSK method for finding the intuitionistic fuzzy optimal solution for fully intuitionistic fuzzy transportation problem in single stage. Also the new multiplication operation on TrIFN is proposed to find the optimal object value in terms of TrIFN. The main advantage of this method is computationally very simple, easy to understand and also the optimum objective value obtained by our method is physically meaningful. Finally the effectiveness of the proposed method is illustrated by means of a numerical example which is followed by graphical representation of the finding.