A Solution Method for Multi-objective Fully Fermatean Fuzzy Transportation Problems Under Stochastic Conditions

Due to their efficient and environmentally friendly features, electric buses have become the key to solving urban transportation and environmental problems. However, existing studies primarily concentrate on optimizing schedules under fixed operational scenarios. This paper proposes a scheduling optimization approach to investigate the impact of stochastic traffic conditions on electric bus operations. First, a static dispatching interval optimization model is established, balancing the interests of the service and operation sides by measuring the passenger and company costs. Then, stochastic model formulas are designed considering three stochastic traffic conditions. Finally, a probabilistic evolution model is developed to improve the genetic algorithm. Based on the bus operation data in Harbin, we validate the model’s performance in enhancing operational benefits. Results reveal that the impact of stochasticity is nonlinear and more significant on passenger travel quality. Furthermore, retrofitting buses and upgrading routes are beneficial to reducing costs and improving travel experience.

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.

Naturally decision makers try to maximize ratio of return/risk, return/cost, and time/cost of transportation problem under uncertain environment. In such cases fractional transportation problem (FTP) play perfect role in instead of ordinary transportation problem. Some limited methods for the solution of fractional transportation problem using exact or fuzzy parameters are available in literature, and existing methods cannot deal uncertain FTP properly having hesitation factors due to inexact information. Here, I introduce the concept of intuitionistic fuzzy expectation of trapezoidal intuitionistic fuzzy numbers, and some related theorems are presented. Further, I present the concept of trapezoidal intuitionistic fuzzy fractional transportation (TIFFTP) problem, and a methodology for the solution of TIFFTP problem based on expectation of trapezoidal intuitionistic fuzzy numbers. Finally, I illustrate the presented TIFFTP problem by using an example.

The single-objective fuzzy or non-fuzzy transportation problem (TP) is not capable of dealing with real-life decision-making problems due to our current competitive market state. In this article, we investigate a fully integer interval multi-objective transportation problem (FIIMOTP) and a fully fuzzy integer multi-objective transportation problem (FFIMOTP). We also provide two solution approaches for solving the FIIMOTP and FFIMOTP. Numerical examples are provided to validate these two approaches. Our results show that the proposed algorithm hugely outperforms the best solution approaches.

A new approach for solving fuzzy transportation problems using generalized trapezoidal fuzzy numbers

In this paper, we introduce a new method to solve Interval-Valued Transportation Problem (IVTP) to deal with those problems of transportation wherein the information available is imprecise. First, a newly proposed fuzzification method is used to convert the IVTP to octagonal fuzzy transportation problem and then with the help of ranking function proposed in this paper, the fuzzy transportation problem is converted into crisp transportation problem. Lastly, Initial Basic Feasible Solution (IBFS) of this problem is obtained using Vogel’s Approximation Method and the solution is improved using Modified Distribution (MODI) method. A numerical example with interval data is solved using the proposed algorithm to make comparison of the solution with some other methods. Also, a numerical example with parameters in the form of octagonal fuzzy numbers is illustrated to compare the effectiveness of the proposed ranking technique. The proposed fuzzification and ranking technique can be used in the other fields of decision making dealing with the data in the same form as considered in this paper.

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 this paper, we establish the fuzzy fractional transportation problem with compensatory approach. To illustrate the proposed compensatory approach to fuzzy fractional transportation problem by using Werner’s ‘fuzzyor’ (μor) operator. We construct optimal solution by using fuzzy Vogel’s approximation method. A numerical example has been provided to illustrate the problem.

In this chapter we will analyze the well known and often employed method for finding the fuzzy optimal solution of fully fuzzy solid transportation problems proposed by Liu and Kao (Eur J Oper Res 153:661–674 2004, [5]). This method can exhibit some shortcomings and that is why we two new, improved methods are proposed in this chapter for solving fully fuzzy solid transportation problems.

In this chapter, a new method (named as Mehar method-II) is proposed to check the existence of at least one more-for-less solution as well as to find all more-for-less solutions (if exists) of symmetric triangular fuzzy transportation problems with mixed constraints (transportation problems with mixed constraints in which each parameter and each variable is represented by a symmetric triangular fuzzy number). Also, to illustrate the proposed Mehar method-II, an existing crisp transportation problem with mixed constraints and an existing symmetric triangular fuzzy transportation problem with mixed constraints are solved by the proposed Mehar method-II. Furthermore, the superiority of the proposed Mehar method-II over some existing methods is discussed.

In the present scenario, transportation model plays a crucial role in supply chain management that reduce the cost and improve the services as per user requirement. There are several methods to find the optimal solution like Vogel’s Approximation Method (VAM), Row minima, Column minima, North-West Corner method etc. In this paper, the proposed method gives the optimal solution and takes less number of iterations for a fuzzy transportation problem as compare to VAM. The Proposed method is illustrated for the secure data transfer from source to the destination node which is considered as Laptop/ Desktop/ Mobile Devices/ Hand-held Devices. Keywords: Fuzzy Transportation; North-West corner method; Optimal Solution; Supply Chain Management.

Chapter 18 - Mehar method for solving unbalanced generalized interval-valued trapezoidal fuzzy number transportation problems

This research uses trapezoidal fuzzy numbers to solve a fuzzy transportation problem in a fuzzy environment. Applied mathematics has extensively researched the transportation problem, which involves solving a set of linear equations with various scientific applications. Simultaneous mathematical models play a crucial role in operational research. This research paper's primary goal is to determine the lowest possible transportation cost for certain supplies via a capacitated network, where the demand and availability notes are previously represented by fuzzy trapezoidal values. The membership function is demonstrated by using trapezoidal fuzzy numbers. This approach is demonstrated by solving a fuzzy transportation problem with trapezoidal fuzzy numbers and the ranking strategy. A numerical example is provided to illustrate that it is the best conceivable solution.

In any decision-making process, imprecision is a significant issue.To deal with the ambiguous environment of collective decision-making, various tools and approaches have been created.Fuzzy set theory is one of the most recent approaches for coping with imprecision.The Fuzzy Transportation Problem (FTP) is a well-known network planned linear programming problem which exists in a variety of situations and has received a lot of attention recently.Many authors defined and solved the fuzzy transportation problem with frequently utilized fuzzy numbers such as triangular fuzzy numbers or trapezoidal fuzzy numbers.On the other hand, real-world problems usually involve more than four variables.To tackle these concerns, the pentagonal fuzzy number is applied to the problems.This article proposes an approach to solving transportation problems whose parameters are pentagonal fuzzy numbers without requiring an initial feasible solution.An algorithm based on the core and spread method and an extended MODI method is developed to determine the optimal solution to the problem.The proposed process is based on the approximation method and gives a more efficient result.An illustrated example is used to validate the model.As a result, the proposed methodology is both simpler and more computationally efficient than the existing approaches.

Manufacturing of steel involves thermal energy intensive processes with coal as the major input. Energy generated is a direct function of ash content of coal and as such it weighs very high as regards the choice of coal. In this paper, we study a multiobjective transportation problem to introduce a new type of coal in a steel manufacturing unit in India. The use of new type of coal serves three non-prioritized objectives, viz. minimization of the total freight cost, the transportation time and the ratio of ash content to the production of hot metal. It has been observed from the past data that the supply and demand points have shown fluctuations around their estimated values because of changing economic conditions. To deal with uncertainties of supply and demand parameters, we transform the past data pertaining to the amount of supply of the ith supply point and the amount of demand of the jth demand point using level (λ,ρ) interval-valued fuzzy numbers. We use a linear ranking function to defuzzify the fuzzy transportation problem. A transportation algorithm is developed to find the non-dominated solutions for the defuzzified problem. The application of the algorithm is illustrated by numerical examples constructed from the data provided by the manufacturing unit.