A new Fermatean fuzzy multi‐objective indefinite quadratic transportation problem with an application to sustainable transportation

In this research work, a soft computing optimization operating approach is developed for a multi-objective aspirational level fractional transportation problem. In the proposed technique, a mathematical model is formulated for the multi-objective aspirational level fractional transportation problem (MOFTP) based on the highest value of one and all objectives of the model. We also used the symmetry concept over our model to identify the best optimum solution based on symmetrical data. We constructed the membership grades for the set of fetched parameters having symmetry. In this work, we also used the concept of ranking function in our mathematical model to obtain the optimum solution of the fuzzy multi-objective fractional transportation. In this proposed algorithm, the aspiration levels are also associated with the objective function of MOFTP. We are also proposing a new approach for the optimization of fractional problems in which the objectives are being optimized by using the numerator function and denominator function simultaneously. Further, a methodology is also developed to find the average cost for each fractional objective of the model. After that, we will find the ranking function for each parameter by using the defuzzification method. By this methodology, we will be able to convert the MOFTP into a bi-objective transportation problem. The provided technique is elaborated with the help of numerical computations to prove the beauty and power of the proposed technique.

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.

Recently, Ghanbari and Mahdavi-Amiri (Appl Math Model 34:3363–3375, 2010) gave the general compromised solution of an LR fuzzy linear system using ABS algorithm. Here, using this general solution, we solve quadratic programming problems with fuzzy LR variables. We convert fuzzy quadratic programming problem to a crisp quadratic problem by using general solution of fuzzy linear system. By using this method, the crisp optimization problem has fewer variables in comparison with other methods, specially when rank of the coefficient matrix is full. Thus, solving the fuzzy quadratic programming problem by using our proposed method is computationally easier than the solving fuzzy quadratic programming problem by using ranking function. Also, we study the fuzzy quadratic programming problem with symmetric variables. We show that, in this case, the associate quadratic programming problem is a convex problem, and thus, we able to find the global optimal.

Transportation problem is an integral part of this competitive world. Minimizing transportation cost of a product from sources to destinations is the main objective of traditional transportation problems. In the present context single objective transportation problem fails to meet the managerial decision making requirements which demands the multi objective transportation problems. Many real life problems can be formulated as multi objective transportation problem, and their solution requires consideration of conflicting objectives. Naturally, these problems have a number of potentially Pareto-optimal solutions. Multi-objective transportation problem refers to a special class of vector minimum linear programming problem, which usually involves multiple, conflicting, and incommensurable objective functions. Solving these complexities, the decision maker has to develop a utility function. In this paper, we develop a utility function to attain the targeted goal for a multi objective transportation problem. This utility function is developed using fuzzy membership functions considering deviations of objective function from the goal value where deviation variable has chosen as fuzzy type. Moreover, fuzzy max-min operator has been included for taking limiting value of the objective functions. Performance of the utility function to achieve proper goal is evaluated and justified by solving numerical examples using fuzzy goal programming.

Transportation problems are a crucial area in logistics and operations research that focuses on effectively transporting products and resources from origin to destination. The complex nature of real-world situations requires the utilization of advanced mathematical models and optimization approaches to formulate effective solutions. Quantifying the accurate supply, demand, and transportation costs of transportation problems is highly challenging due to the market’s unpredictable economic and environmental conditions. Several methodologies have been proposed to address the issue of transportation problems with uncertain parameters, utilizing fuzzy theory and its derivatives. Complex Pythagorean fuzzy numbers (CPyFNs) are highly suitable and effective for representing uncertain and ambiguous information in real-world scenarios. This article introduces an innovative analysis of transportation problems in the complex Pythagorean fuzzy information framework. Initially,we establish an improved score function value to evaluate the ordering of CPyFNs more precisely and propose several ranking rules. The following section of the study presents mathematical formulations and optimization models for transportation problems, where the parameters are expressed as CPyFNs. The solution approach is formulated using Vogel’s approximation method (VAM) and modified distribution method (MODI) to solve transportation problems in a complex Pythagorean fuzzy setting. An illustrative numerical example is provided to showcase the practicality and effectiveness of the proposed method in real-life scenarios. The significance of the research and the extent of future investigation are also emphasized.

This article, in order to address impreciseness, initiated the notion of dual hesitant fermatean fuzzy sets (DHFFSs), as a generalization of the combination of dual hesitant fuzzy set (DHFS), dual hesitant Pythagorean fuzzy set (DHPFS) and Fermatean fuzzy set (FFS). The authors defined the fundamental set of operations for DHFFS. Additionally, the authors have also proposed two ranking functions and an accuracy function for the ordering of this novel set. In order to facilitate the pragmatic implementation of DHFFS in optimization, the authors formulated three types of transportation problem with dual hesitant Fermatean fuzzy (DHFF) parameters. To optimize the DHFF-TP, an algorithm was proposed with the help of one of the proposed ranking functions. Artificial neural network is also applied to the transportation problems in DHFF environment. A numerical example based on the transportation of COVID-19 vaccine with DHFF cost has also been carried out to validate out to validate our technique. In this article the proposed fuzzy system is included into this model to control uncertainties in Time, Quality, Cost, Reliability, and Availability induced in the transportation interruptions. The paper illustrates implication of the proposed fuzzy based model using a simulation in the context of a transportation process. The proposed model seeks to satisfy customers' demands for high-quality products and services, on a timely manner, at lowest cost possible. These characteristics are essential during times of disturbance, such as the COVID-19 pandemic.

Many problems appear to be arising in the present as a result of variations in transportation networks. The stochastic fixed-charge transportation problem (SFCTP) is one such problem. The SFCTP is transformed into a chance-constrained programming (CCP) problem where supply and demand are stochastic and objective functions are in a rough interval. In this model, to analyze the multi-objective rough interval stochastic fixed-charge transportation problem (MORISFCTP), where the objective function coefficients are represented by rough intervals and the supply and destination factors are probabilistic constraints. This model operates an expected value operator to deal with uncertainty, in which the coefficient of the objective functions in the fuzzy is changed to a crisp form, and the probabilistic constraints are converted to a deterministic form by the Weibull distribution. To produce the optimal compromise solutions of the proposed model, three distinct methods are used: the fuzzy programming approach, the method of a linear weighted sum, and the €-constraint method. Lastly, the paper delivers a practical illustration of a MORISFCTP to demonstrate the usefulness and feasibility of the suggested methodology.

This paper is to integrate among solid transportation problem, budget constraints and carbon emission with probable maximum profit. The range of air pollution and climate change are effected by \({\text {CO}_2}\) gas emission and other greenhouse gases from myriad transportation systems. Henceforth, our target is to minimize carbon emission for pollution-free environment. Transportation system with single objective is hardly applicable to the situation with more than one criterion. Therefore multi-objective decision making is incorporated for designing real-life transportation problem. Due to time pressure, data limitation, lack of information or measurement errors in practical problems, there exist some hesitations. Based on the fact, decision maker considers indeterminacy in the designed problems. To overcome such hesitancy of occurrence and non-occurrence in fuzzy and intuitionistic fuzzy, neutrosophic set is very important and suitable to accommodate such general structure of problems. Therefore neutrosophic environment with neutrosophic linear programming, fuzzy programming and global criterion method is utilized here to find the compromise solution of the multi-objective transportation problem. Thereafter, the performance of the considered model is useful by evaluating a numerical example, and then the derived results are compared. Finally sensitivity analysis and conclusions with upcoming works of this research are stated hereafter.

Sustainability encompasses the interaction between environmental, economic, and social systems at various levels. In transportation systems, vehicles rely on different fuels to meet energy demands. However, these fuels contribute to environmental pollution by releasing greenhouse gases. Considering this context, the present study investigates a sustainable fixed-charge multi-objective multi-driver multi-route multi-conveyance shortest path problem (FCMOMDMRMCSPP). The primary intent is to identify the most efficient route that minimizes cost, time, risk, carbon emissions, and distance while traveling between sources and destinations. Real-world scenarios often involve uncertainties, making it difficult to precisely define the parameters of FCMOMDMRMCSPP. Triangular fuzzy numbers are introduced in the proposed model to address this ambiguity. The possibilistic programming approach is employed to transform the fuzzy problem into a deterministic one. The resulting deterministic model is then solved using two multi-objective genetic algorithms (MOGAs): aspiration level (AL)-based non-dominated sorting genetic algorithm (NSGA)-II and NSGA-III. An aquatic fish haul system problem has been examined as an application within the framework of the proposed model. For α = 0 and 0.1, the AL-based NSGA-II and III generate 6, 11, and 4 solutions that satisfy the decision-maker’s (DM’s) ALs in cases I, II, and III of the AL and shape parameter combination, respectively. Similarly, for α = 0.5, these methods yield 5, 11, and 4 solutions, while for α = 0.9, they provide 6, 11, and 4 solutions for the same cases. A comparative analysis is conducted between the compromise solutions obtained from the proposed techniques and those derived from the hybrid genetic algorithm, NSGA-II and NSGA-III. A sensitivity analysis of the objective functions concerning the shape parameter and AL is also performed. Finally, the coverage performance measure is computed to assess the effectiveness of the proposed methods.

Solution of Multi-Objective Interval Solid Transportation Problems Using Expected Value     International Journal of Computer & Organization Trends  (IJCOT)             © 2014 by IJCOT

Cost-time Minimization in a Transportation Problem with Fuzzy Parameters: A Case Study

Recently, ranking data with respect to the intrinsic geometric structure (manifold ranking) has received considerable attentions, with encouraging performance in many applications in pattern recognition, information retrieval and recommendation systems. Most of the existing manifold ranking methods focus on learning a ranking function that varies smoothly along the data manifold. However, beyond smoothness, a desirable ranking function should vary monotonically along the geodesics of the data manifold, such that the ranking order along the geodesics is preserved. In this article, we aim to learn a ranking function that varies linearly and therefore monotonically along the geodesics of the data manifold. Recent theoretical work shows that the gradient field of a linear function on the manifold has to be a parallel vector field. Therefore, we propose a novel ranking algorithm on the data manifolds, called Parallel Field Ranking. Specifically, we try to learn a ranking function and a vector field simultaneously. We require the vector field to be close to the gradient field of the ranking function, and the vector field to be as parallel as possible. Moreover, we require the value of the ranking function at the query point to be the highest, and then decrease linearly along the manifold. Experimental results on both synthetic data and real data demonstrate the effectiveness of our proposed algorithm.

In this study, we consider solution approaches used to solve the proposed bilevel linear programming model for an Industrial Symbiosis network. We first solve this model with the well-known Karush-Kuch-Tucker (KKT) approach. However, transforming a bilevel programming model with the KKT approach increases the number of variables and constraints. For this reason, we use the fuzzy programming approach and the fuzzy goal programming approaches as alternatives to the KKT approach. Next, we compare the results of the KKT approach with these methods and examine the suitability of these approaches to solve our bilevel problem. Unlike previous studies, which claimed that fuzzy approaches mostly outperform the KKT approach, in our case, the best solution is obtained with the KKT approach. This is most probably because these approaches ignore the hierarchical nature of the problem. We believe that more research on fuzzy approaches is needed to evaluate the suitability of these approaches for solving bilevel programming problems.KeywordsIndustrial symbiosisBilevel programmingFuzzy programming approachFuzzy goal programming

There are various studies in which a variety of prediction tools have been introduced in time-series prediction literature. Non-probabilistic approaches which are based on fuzzy set theory, especially in recent years, have been put forward. Although these approaches including adaptive network fuzzy inference system, fuzzy functions approach, and fuzzy regression can be successfully utilized as a prediction tool, they have not been designed for prediction problem and they pass over the dependency structure of time-series observations. From this point forth, designing a prediction tool that considers the dependency structure of the observations of time series will procure to get predictions more accurately. Although the membership values, in the analysis process, are taken into account in almost all fuzzy methods, the non-membership and hesitation values are not considered. However, using as much information as possible on time series may be another positive factor that gives more accurate predictions. The primary aim of this study, for time-series prediction, is to introduce an intuitionistic fuzzy regression functions approach based on hesitation margin (I-FRF-HM). In the introduced intuitionistic fuzzy regression functions approach, two inference systems are separately constituted such that while one of them uses membership, other one uses non-membership values as inputs of inference system in addition with the crisp observations of time series. Predictions obtained from each system are converted into final predictions of whole inference system via an approach based on hesitation margin. Intuitionistic fuzzy C-means are utilized to get membership and non-membership values in the proposed model. The proposed I-FRF-HM has been applied to various real-world time series. The obtained findings are evaluated along with the results of some other time-series prediction models. The results show that the proposed I-FRF-HM has superior prediction performance to other prediction models.

The problem of transport is a special type of mathematical programming designed to search for the optimal distribution network, taking into account the set of suppliers and the set of recipients. This article proposes an innovative approach to solving the transportation problem and devises source codes in GNU Octave (version 3.4.3) to avoid the necessity of carrying out enormous calculations in traditional methods and to minimize transportation costs, fuel consumption, and CO2 emission. The paper presents a numerical example of a solution to the transportation problem using: the northwest corner, the least cost in the matrix, the row minimum, and Vogel’s Approximation Methods (VAM). The joint use of mathematical programming and optimization was applicable to real conditions. The transport was carried out with medium load trucks. Both suppliers and recipients of materials were located geographically within the territory of the Republic of Poland. The presented model was supported by a numerical example with interpretation and visualization of the obtained results. The implementation of the proposed solution enables the user to develop an optimal transport plan for individually defined criteria.