More details on this study can be found in my recent article "Algorithms for solving the optimization problems using fuzzy and intuitionistic fuzzy set" published in International Journal of System Assurance Engineering and Management (https://doi.org/10.1007/s13198-019-00941-3).

Transportation problems (TPs) and assignment problems (APs) are optimization problems. The objective of the optimization problem is to minimize/maximize the objective function subject to the given constraints, including non-negativity restrictions. Solving APs and TPs in a crisp environment, we have several algorithms. But, in real life, the AP's parameter (time/cost) and TP's parameters (cost, demand, supply) are not fixed numbers. For example, the transportation cost is not a fixed crisp number; it may vary due to variations in the rates of petrol or diesel (i.e., it is a fuzzy/intuitionistic fuzzy number). As a result, we cannot use conventional methods to solve the problem. So, this article presents a novel and efficient fuzzy and intuitionistic fuzzy sets-based algorithm for solving optimization problems in three distinct environments, along with an amazing theorem and some new and tangible results.

It effectively solves the following issues with the new algorithm and software:

1. FTP: The existing book chapter solves the problem using the PSK method, but this article solves it with TORA and the new algorithm.
2. IFTP: The existing research article solves the problem using the PSK method, but this article solves it with TORA and the new algorithm.
3. IFSTP: The existing article solves the type-1 IFSTP, but this article solves the type-2 IFSTP with TORA and the new algorithm.
4. FAP: The existing research article gives ideas to solve the FAP using the PSK method, but this article directly solves the FAP with TORA and the new algorithm.
5. IFAP (minimization case): The existing article solves the problem using the PSK method, but this article solves it with TORA and the new algorithm.
6. IFAP (maximization case): The existing Ph.D. thesis solves the problem using the PSK method, but this article solves it with software and the new algorithm.
7. IFSAP: The existing article solves the IFSAP using Lingo 17.0, but this article solves the IFSAP with TORA and the new algorithm.

Crisp, fuzzy, and intuitionistic fuzzy optimization problems are widely used in organizations, coal transportation, timetabling issues, and other fields. Thus, studying optimization problems in three different environments with one algorithm is among the most interesting in mathematical history. Therefore, in the literature, a number of mathematicians have proposed several algorithms to deal with these problems^1,2,3. However, this article deals with all issues in a single algorithm.

Because the parameters of many real-world optimization problems are uncertain, they can be challenging to solve. To solve optimization problems with uncertain data, we have many fuzzy set-based algorithms in the literature (the reliability and accuracy of the solution are the question mark)⁴. However, the fuzzy set only takes membership values into account, ignoring the hesitation index and non-membership values^5,6. It is the fuzzy set's main drawback. As a result, the author of this study considers optimization problems that include both fuzzy and intuitionistic fuzzy numbers in their parameters. The author delineates the problems and employs TrFN, TIFNs, TrIFN, and TFN to address them. The author offers a new software-based approach for finding the optimal solution to the proposed optimization problems. Additionally, the fuzzy and intuitionistic fuzzy optimization problems and their conventional problem solutions are obtained simultaneously. In everyday business and organizations, intuitionistic fuzzy optimization problems are used extensively. Therefore, this post will help everyone find the article, and the original article will be helpful to everyone who reads it and draw attention from the general public.

The main advantages of this algorithm are as follows:

1. It solves a variety of optimization problems with a single algorithm.
2. It can also be useful in solving complex problems.
3. It is independent of existing crisp, fuzzy and intuitionistic fuzzy methodologies.
4. It saves the decision-maker's computational time and yields an optimal solution in fewer steps.
5. It helps to solve fuzzy, crisp, intuitionistic fuzzy, balanced, maximization, minimization, and unbalanced optimization problems.
6. It provides ideas for performing sensitivity analysis and is very simple computationally.
7. This algorithm always yields a physically meaningful, reliable, and appropriate solution, i.e., the optimal solution, because the author has proved the necessary theorem (i.e., the PSK theorem).

In the history of mathematics, the author P. Senthil Kumar (PSK) solved optimization problems (TPs and APs) under three distinct environments. He originally divided FTPs into four categories (types 1, 2, 3, and 4) and IFSAPs into two categories (mixed and full IFSAPs)^7,8,9. He solved all problems using various software methods, including the PSK method, with the help of the PSK theorem. This article deals with optimization problems in three distinct environments. Studying optimization problems in three distinct environments is better than studying optimization problems in a crisp environment. Since the parameters of the optimization problems are not fixed numbers. For more details, see here^10,11,12.

In our daily lives, there is a need to allocate resources economically. Understanding the optimization problem is critical to effectively managing this situation. However, because of unforeseen environments, the parameters of optimization problems are not at all crisp numbers^13,14,15. As a result, resolving optimization problems with three distinct environments is more critical. So, this work effectively resolves the optimization problems in crisp, fuzzy, and intuitionistic fuzzy environments. We can study similar issues with different approaches and limits here^16,17. Fig. 1 shows the graphical representation of the trapezoidal intuitionistic fuzzy number (TrIFN).

Fig. 1 Graphical Representation of TrIFN

The TORA output summary for type-2 IFTP with TrIFNs is given in Fig. 2.

Fig. 2 TORA output summary for type-2 IFTP with TrIFNs

References:

1. Kumar, P.S. A simple method for solving type-2 and type-4 fuzzy transportation problems. Int J Fuzzy Logic Intell Syst 16(4):225–237, https://doi.org/10.5391/IJFIS.2016.16.4.225, (2016).
2. Kumar, P.S, Hussain, R.J. A systematic approach for solving mixed intuitionistic fuzzy transportation problems. Int J Pure Appl Math 92:181–190, https://doi.org/10.12732/ijpam.v92i2.4, (2014).
3. Kumar, P. S. Developing a new approach to solve solid assignment problems under intuitionistic fuzzy environment. Int J Fuzzy Syst Appl 9(1):1–34, https://doi.org/10.4018/ijfsa.2020010101, (2020).
4. Kumar, P.S. Search for an optimal solution to vague traffic problems using the PSK method. Handbook of Research on Investigations in Artificial Life Research and Development, IGI Global, pp. 219–257, https://doi.org/10.4018/978-1-5225-5396-0.ch011, (2018).
5. Kumar, P.S. A note on ‘a new approach for solving intuitionistic fuzzy transportation problem of type-2’. Int J Logist Syst Manag 29(1):102–129, https://doi.org/10.1504/ijlsm.2018.088586; https://doi.org/10.1504/ijlsm.2018.10009204, (2018).
6. Kumar, P. S. Ai-driven decision support system for intuitionistic fuzzy assignment problems. Using Traditional Design Methods to Enhance AI-Driven Decision Making, IGI Global, pp. 352–398, https://doi.org/10.4018/979-8-3693-0639-0.ch016, (2024).
7. Kumar, P. S. Linear programming approach for solving balanced and unbalanced intuitionistic fuzzy transportation problems. Int J Oper Res Inf Syst 9(2):73–100, https://doi.org/10.4018/ijoris.2018040104, (2018).
8. Hussain, R.J, Kumar, P.S. An optimal more-for-less solution of mixed constraints intuitionistic fuzzy transportation problems. Int J Contemp Math Sci 8:565–576, https://doi.org/10.12988/ijcms.2013.13056, (2013).
9. Kumar, P. S. The PSK method for solving fully intuitionistic fuzzy assignment problems with some software tools. Theoretical and Applied Mathematics in International Business, IGI Global, pp. 149–202, https://doi.org/10.4018/978-1-5225-8458-2.ch009, (2020).
10. Kumar, P.S. The PSK Method: A new and efficient approach to solving fuzzy transportation problems. Transport and Logistics Planning and Optimization, IGI Global, pp. 149–197, https://doi.org/10.4018/978-1-6684-8474-6.ch007, (2023).
11. 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).
12. Kumar, P.S. Computationally simple and efficient method for solving real-life mixed intuitionistic fuzzy 3D assignment problems. In J Softw Sci Comput Intell 14(1):1–42, https://doi.org/10.4018/IJSSCI.291715, (2022).
13. Kumar, P. S. Algorithms and software packages for solving transportation problems with intuitionistic fuzzy numbers. Operational Research for Renewable Energy and Sustainable Environments, IGI Global, pp. 1–55, https://doi.org/10.4018/978-1-6684-9130-0.ch001, (2023).
14. Kumar, P. S. Finding the solution of balanced and unbalanced intuitionistic fuzzy transportation problems by using different methods with some software packages. Handbook of Research on Applied AI for International Business and Marketing Applications, IGI Global, pp. 278–320, https://doi.org/10.4018/978-1-7998-5077-9.ch015, (2021).
15. 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).
16. 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).
17. Kumar, P.S. An efficient approach for solving type-2 intuitionistic fuzzy solid transportation problems with their equivalent crisp solid transportation problems. Int J Syst Assur Eng Manag 15(9):4370–4403, https://link.springer.com/10.1007/s13198-024-02433-5, (2024).