More details on this study can be found in our recent article "An efficient approach for solving type-2 intuitionistic fuzzy solid transportation problems with their equivalent crisp solid transportation problems" published in International Journal of System Assurance Engineering and Management (https://doi.org/10.1007/s13198-024-02433-5).

Solving the 3-dimensional type-2 intuitionistic fuzzy transportation problem (type-2 IFSTP) in a real-world situation with software codes is new in the literature. Not only the theorem, corollary, and their proof, as well as some results relevant to type-2 IFSTPs, are presented, but also the existing problem's solution, and the proposed solution by the different software for the same problem, are analyzed in this article. Shortcomings in previous results are effectively addressed; also, the proper solution is obtained for both the solid transportation problem (STP) and type-2 IFSTP using a single LP model, which is the primary goal of this research study. Further, it discusses the sensitivity analysis, time complexity of the existing algorithms for solving STPs with different intuitionistic fuzzy numbers (IFNs), and the benefits of the proposed methodology. In real life, the type-2 IFSTP has wide applications in industry, rocket launching, telecommunication, coal transportation, et cetera¹. So, there is no doubt that it will attract researchers, scientists, governments, academicians, policymakers, business executives, master's degree students, and practitioners.

In several real-life situations, there is a need to transport goods from different factories to different warehouses². Studying the 3-dimensional transportation problem is very important to handling this situation effectively³. However, transportation parameters are not real numbers due to some uncontrollable factors. Hence, solving transportation problems with IFNs is more important⁴. So, this study effectively solves the 3-dimensional type-2 IFTP. To read similar types of problems with different constraints, we can see^{5, 6, 7}. Table 1 compares the proposed study with the existing literature.

Table 1 Comparison table

The uniqueness of the research is as follows:

1. Matlab-based IFSTP solution is novel in the literature. It also includes Matlab code for type-2 IFSTPs with graphical solution representations.
2. Shortcomings in Pandian and Anuradha's (2010) study were identified and addressed.
3. This paper aims to achieve optimal solutions and objective values for both the STP and type-2 IFSTPs using a single LP model.
4. The time complexity of current algorithms for solving STPs with IFNs has been investigated.
5. It checks the theoretical result against two alternative software simulation findings. Furthermore, these applications and programs are useful for doing sensitivity studies in type-2 IFSTPs.
6. This article provides a new theorem and a corollary for type-2 IFSTPs, along with some conventional results. It also demonstrates that the answer found using the provided method/algorithm is always viable and optimal.

The block diagram (see Fig. 1) shows the steps for the presented algorithm.

Fig. 1 Logical steps for the algorithm

The solutions for the proposed problem are shown through 2-D (see Fig. 2) and 3-D (see Fig. 3) plots using Matlab.

Fig. 2 2-Dimensional plot for the solution of example No. 3

Fig. 3 3-Dimensional plot for the solution of example No. 3

The concept of intuitionistic fuzzy number is not easily help for the decision maker to solve the transportation problem. How an expert will provide such ratings during the problems? To answer this question, the author has presented real-life examples in this article. This article also presents accurate solutions to many scientific questions with the PSK theorem in optimization problems. The recent advancement in this field can be found here^{8, 9, 10}.

The author does not mention type-2 intuitionistic fuzzy logic anywhere in this paper. The author's future research plans include solving type-2 IFSTPs with type-2 intuitionistic fuzzy logic or other intuitionistic fuzzy set extensions. The open problem is also included in the conclusion section.

References:

1. Kumar, P.S. PSK method for solving intuitionistic fuzzy solid transportation problems. Int J Fuzzy Syst Appl 7(4):62–99, https://doi.org/10.4018/ijfsa.2018100104, (2018).
2. Kumar, P.S, Hussain, R.J. Computationally simple approach for solving fully intuitionistic fuzzy real life transportation problems. Int J Syst Assur Eng Manag 7:90–101, https://link.springer.com/article/10.1007/s13198-014-0334-2, (2016).
3. Kumar, P.S. Algorithms for solving the optimization problems using fuzzy and intuitionistic fuzzy set. Int J Syst Assur Eng Manag 11(1):189–222, https://link.springer.com/10.1007/s13198-019-00941-3, (2020).
4. Kumar, P.S. Intuitionistic fuzzy zero point method for solving type-2 intuitionistic fuzzy transportation problem. Int J Oper Res 37(3):418–451, https://doi.org/10.1504/ijor.2020.105446, https://doi.org/10.1504/ijor.2020.10027072, (2020).
5. Kumar, P.S. Intuitionistic fuzzy solid assignment problems: a software-based approach. Int J Syst Assur Eng Manag 10(4):661–675, https://link.springer.com/article/10.1007/s13198-019-00794-w, (2019).
6. Kumar, P.S. Computationally simple and efficient method for solving real-life mixed intuitionistic fuzzy 3D assignment problems. In J Softw Sci Comput Intell (IJSSCI) 14(1):1–42, https://doi.org/10.4018/IJSSCI.291715, (2022).
7. Kumar, P. S. Theory and applications of the software-based PSK method for solving intuitionistic fuzzy solid assignment problems. Applications of New Technology in Operations and Supply Chain Management, IGI Global, pp. 360–403, https://doi.org/10.4018/979-8-3693-1578-1.ch019, (2024).
8. Kumar, P. S. The theory and applications of the software-based PSK method for solving intuitionistic fuzzy solid transportation problems. Perspectives and Considerations on the Evolution of Smart Systems, IGI Global, pp. 137–186, https://doi.org/10.4018/978-1-6684-7684-0.ch007, (2023).
9. Kumar, P. S. PSK method for solving mixed and type-4 intuitionistic fuzzy solid transportation problems. Int J Oper Res Inf Syst 10(2):20–53, https://doi.org/10.4018/ijoris.2019040102, (2019).
10. Kumar, P. S. A simple and efficient algorithm for solving type-1 intuitionistic fuzzy solid transportation problems. Int J Oper Res Inf Syst 9(3):90–122, https://doi.org/10.4018/ijoris.2018070105, (2018).