Multi-objective Transportation Problem Research Articles

Multi-objective optimization model for railway heavy-haul traffic: Addressing carbon emissions reduction and transport efficiency improvement

This paper establishes a multi-objective optimization model for railway heavy-haul trains, focusing on reducing carbon emissions and improving transport efficiency. The model integrates optimization of the route and the vehicle load rate, significantly reducing carbon emissions and enhancing transport efficiency. It addresses the challenges and characteristics of heavy-haul trains, introducing multi-objective optimization problems related to transport carbon emissions and efficiency. Using a pigeon-inspired optimization algorithm, the model considers joint constraints between carbon emissions and transport efficiency objectives. To overcome challenges in multi-objective transportation problems, the paper proposes a forward-learning pigeon-inspired optimization algorithm based on a surrogate-assisted model. This approach calculates the quality of the candidate solution using a surrogate model, reducing time costs. The algorithm employs a forward-learning strategy to enhance learning from non-dominant solutions. Experimental validation with benchmark functions confirms the effectiveness of the model and offers optimized solutions. The proposed method reduces carbon emissions while maintaining transport efficiency, contributing innovative ideas for the development of sustainable heavy-duty trains.

Solving a multi-choice solid fractional multi objective transportation problem: involving the Newton divided difference interpolation approach

<abstract> <p>Multi-objective transportation problems (MOTPs) are mathematical optimization problems that involve simultaneously considering multiple, often conflicting objectives in transportation planning. Unlike traditional transportation problems, which typically focus on minimizing a single objective such as cost or distance, MOTPs aim to balance multiple objectives to find the optimal solution. These problems appear in various real-world applications such as logistics, supply chain management, and transportation, where decision-makers need to consider multiple criteria when designing transportation networks, routing vehicles, or scheduling deliveries. The primary challenge lies in the uncertainty in real-world transportation scenarios, where logistics involve factors beyond cost and distance. We investigated a multi-choice solid fractional multi-objective transportation problem (MCSF-MOTP) based on supply, demand, and conveyance capacity, where the coefficients of the objective functions were of the multi-choice type due to uncertainty. To address this uncertainty, the proposed model employed the Newton divided difference interpolation polynomial method, and the suitability of this model was validated through a numerical illustration employing a ranking approach.</p> </abstract>

An Effective Approach for Solving Multi-objective Transportation Problem

In this present study, a transportation problem is considered such that the total cost and time of transportation are minimized without taking into account their priorities. In literature, there are less techniques available for finding the efficient solutions of multi-objective transportation problem. So, we developed a heuristic algorithm to find most efficient solution of multi-objective transportation problem, which gives efficient solution with minimum difference from ideal solution. Firstly, we aim at formulating a multi-objective transportation problems along with a novel algorithm to find efficient solutions. The proposed algorithm gives optimal solution faster in comparison to other available techniques in literature for the given multi-objective transportation problem. Moreover, it avoids the degeneracy as well as requires low computational effort. Furthermore, an illustrative example is provided to show the feasibility and applicability of the proposed approach and compare the results with the existing approaches to show the effectiveness of it.

Neutrosophic hyperbolic programming strategy for uncertain multi-objective transportation problem

This paper suggests a neutrosophic hyperbolic programming approach based on the hyperbolic membership function to get an efficient solution for an uncertain multi-objective transportation problem where the uncertain normal distribution presents uncertainty. This technique provides flexibility to the decision-maker to select various confidence levels according to his/her own choice and analyze the satisfaction percentage of the obtained optimal compromise solution along with its truth, indeterminacy, and falsity values. In this approach, the parameters are changed from uncertain to crisp by using the inverse measure theorem, and the neutrosophic non-linear hyperbolic membership functions are used to convert the whole model into a non-linear programming problem in a neutrosophic environment. The advantages of our recommended method in real-world applications are effectively illustrated by a numerical illustration. This illustration also makes it easier to compare the results attained using our suggested strategy to those obtained using the methodologies put forward by Uddin et al. (2021) and Miah et al. (2022). Also, MATLAB software (version R2019a,64bit(win64)) is used to develop the code for the solution of this non-linear uncertain multi-objective transportation problem.

A generalized parametric approach for solving different fuzzy parameter based multi-objective transportation problem

A generalized parametric approach for solving different fuzzy parameter based multi-objective transportation problem

An overview of weighted goal programming: a multi-objective transportation problem with some fresh viewpoints

An overview of weighted goal programming: a multi-objective transportation problem with some fresh viewpoints

Multi-choice fractional stochastic multi-objective transportation problem

Multi-choice fractional stochastic multi-objective transportation problem

A multi-objective transportation problem under quantity dependent credit period and cost structure policies in triangular intuitionistic fuzzy environment

One of the major economic activities of a business concern is transportation of its products without any hassle. Therefore, the research on a transportation problem is very important now-a-days. In a classical transportation problem, the amount incurred for shipping the goods from the source to a destination is determined on the basis of unit transportation cost. But now, in many practical cases, it is seen that the unit transportation cost is settled on the basis of transported amount. In such a direction, this article develops a transportation problem to transport the produced items from a production house to some retailers through some distributors. The system has two cost parameters, one is actual unit transportation cost and other is demanded unit transportation cost. The actual unit transportation cost is the cost incurred during the transportation while the demanded unit transportation cost is the cost demanded by the distributors. Here, the demanded unit transportation cost has been decided depending on the amount of transported item. Since in the system, there are more than one distributor, distributors offer a credit period policy to attract the retailers. On the basis of the deterioration of the loaded amount, the transported items are classified into two categories. For the higher deteriorated items, the transportation cost is relatively larger and for the lower deteriorated items, the transportation cost is relatively lower with the increase in the loading amount. Considering the above two categories, two crisp models are developed. Due to the uncertain information of the actual unit transportation cost, the models are illustrated considering the actual unit transportation cost as a triangular intuitionistic fuzzy number. The main objective of this research is to minimize the total cost of the retailers and to maximize the total profit of the distributors simultaneously under the above circumstances. For crispification, two different methods have been used here — one is an existing method namely ordered weighted average operator and other is a developed average value approach. Also, the elitist non-dominated sorting genetic algorithm (NSGA-II) and the fuzzy programming technique have been used to get the optimal solution of this problem. To justify the impact and significance of the proposed models on this decision making system, some numerical studies have been done and it is observed that distributors and retailers both benefit more under the credit period policy than the policy without credit period. This study discusses the situation under which the uncertain parameters are to be expressed in terms of triangular intuitionistic fuzzy numbers. Also, it is identified that more accurate crispification has been done by the average value approach than the existing approach.

An ε-Constraint Method for Multiobjective Linear Programming in Intuitionistic Fuzzy Environment

Effective decision-making requires well-founded optimization models and algorithms tolerant of real-world uncertainties. In the mid-1980s, intuitionistic fuzzy set theory emerged as another mathematical framework to deal with the uncertainty of subjective judgments and made it possible to represent hesitancy in a decision-making problem. Nowadays, intuitionistic fuzzy multiobjective linear programming (IFMOLP) problems are a topic of extensive research, for which a considerable number of solution approaches are being developed. Among the available solution approaches, ranking function-based approaches stand out for their simplicity to transform these problems into conventional ones. However, these approaches do not always guarantee Pareto optimal solutions. In this study, the concepts of dominance and Pareto optimality are extended to the intuitionistic fuzzy case by using lexicographic criteria for ranking triangular intuitionistic fuzzy numbers (TIFNs). Furthermore, an intuitionistic fuzzy ε-constraint method is proposed to solve IFMOLP problems with TIFNs. The proposed method is illustrated by solving two intuitionistic fuzzy transportation problems addressed in two studies (S. Mahajan and S. K. Gupta’s, “On fully intuitionistic fuzzy multiobjective transportation problems using different membership functions,” Ann Oper Res, vol. 296, no. 1, pp. 211–241, 2021, and Ghosh et al.’s, “Multi-objective fully intuitionistic fuzzy fixed-charge solid transportation problem,” Complex Intell Syst, vol. 7, no. 2, pp. 1009–1023, 2021). Results show that, in contrast with Mahajan and Gupta’s and Ghosh et al.’s methods, the proposed method guarantees Pareto optimality and also makes it possible to obtain multiple solutions to the problems.

The Multi-Objective Transportation Problem Solve with Geometric Mean and Penalty Methods

The traditional (classical) Transportation Problem (TP) can be viewed as a specific case of the Linear Programming (LP) problem, as well as its models are used to find the best solution for the problem of predetermined how many units of a good or service need to be shipped from one source to multiple locations, with the goals being to reduce time or expense. Classical TP has one objective but when there are two or more objectives to be optimized for in a TP, the strategies used to optimize a single objective are inapplicable. The term “Multi-Objective Transportation Problem (MOTP)” refers to situations in which there are two or more objectives in a TP. The specific extension of the transportation problem is the MOTP. This work provided a novel alternative algorithm that uses geometric means along with the penalty technique to address MOTP. Specifically, analyzed data by comparing our method with numerical examples and presenting the results in a line graph. Our analysis shows that our approach yields better solutions than existing methods, demonstrating the novelty and effectiveness of our approach. The comparison with numerical examples provides a clear and intuitive way of presenting the superiority of our method, making it accessible to practitioners and researchers in the field. These findings have important implications for improving the accuracy and reliability of solutions to the problem at hand. Overall, our study contributes to the advancement of the field by providing a novel and effective method for solving the problem.

To Use an Ant Colony Technique to Solve a Crispy Type Bi- and Tri-Objective Transportation Problem

Transportation problem (TP) aims to reduce the entire transportation cost of moving resources from various supply hubs to various demand hubs. However, in real-life situations, all organizations want to achieve numerous objectives while making transportation of goods. The degree of deterioration may vary depending on the mode, route, and time of transport. In some cases, the multiobjective could be used to reduce the use of a scarce resource, such as energy. As a result, the proposed approach was discovered to be an algorithm that improves bi- and triobjective TP techniques. This is an innovative way for solving the new bi- and triobjective transportation algorithm using a modified ant colony optimization (ACO) algorithm. According to the literature, various strategies have been developed in the past to tackle the multiobjective transportation problem (MOTP). The MOTP is solved using goal programming, fuzzy programming, interactive solution algorithms, and other techniques. These strategies are occasionally good or bad in achieving better results in a reasonable amount of time. The heuristic technique used in this work is the improved ACO algorithm, which is based on the ant colony algorithm and has been found to provide solutions with a reasonable degree of satisfaction for two and three objective TPs. When the findings are compared, the solution achieved using the proposed method has delivered the best performance and provides a case study to show the new strategy.

On neutrosophic multi-level multi-objective linear programming problem with application in transportation problem

In this research, we use the harmonic mean technique to present an interactive strategy for addressing neutrosophic multi-level multi-objective linear programming (NMMLP) problems. The coefficients of the objective functions of level decision makers and constraints are represented by neutrosophic numbers. By using the interval programming technique, the NMMLP problem is transformed into two crisp MMLP problems, one of these problems is an MMLP problem with all of its coefficients being upper approximations of neutrosophic numbers, while the other is an MMLP problem with all of its coefficients being lower approximations of neutrosophic numbers. The harmonic mean method is then used to combine the many objectives of each crisp problem into a single objective. Then, a preferred solution for NMMLP problems is obtained by solving the single-objective linear programming problem. An application of our research problem is how to determine the optimality the cost of multi-objective transportation problem with neutrosophic environment. To demonstrate the proposed strategies, numerical examples are solved.

Fermatean Fuzzy Programming with New Score Function: A New Methodology to Multi-Objective Transportation Problems

The aim of this work is to establish a new methodology to tackle the multi-objective transportation problems [MOTP] in a Fermatean fuzzy environment that can deal with all the parameters that possess a conflicting nature. In our research work, we developed a new score function in the context of a fermatean nature for converting fuzzy data into crisp data with the help of the Fermatean fuzzy technique. Then, we introduced an algorithm-based methodology, i.e., the Fermatean Fuzzy Programming approach to tackle transportation problems with multi-objectives. The main purpose of this research work is to give an alternate fuzzy programming approach to handle the MOTP. To justify the potential and validity of our work, numerical computations have been carried out using our proposed methodology.

Extended DEA method for solving multi-objective transportation problem with Fermatean fuzzy sets

<abstract><p>Data envelopment analysis (DEA) is a linear programming approach used to determine the relative efficiencies of multiple decision-making units (DMUs). A transportation problem (TP) is a special type of linear programming problem (LPP) which is used to minimize the total transportation cost or maximize the total transportation profit of transporting a product from multiple sources to multiple destinations. Because of the connection between the multi-objective TP (MOTP) and DEA, DEA-based techniques are more often used to handle practical TPs. The objective of this work is to investigate the TP with Fermatean fuzzy costs in the presence of numerous conflicting objectives. In particular, a Fermatean fuzzy DEA (FFDEA) method is proposed to solve the Fermatean fuzzy MOTP (FFMOTP). In this regard, every arc in FFMOTP is considered a DMU. Additionally, those objective functions that should be maximized will be used to define the outputs of DMUs, while those that should be minimized will be used to define the inputs of DMUs. As a consequence, two different Fermatean fuzzy effciency scores (FFESs) will be obtained for every arc by solving the FFDEA models. Therefore, unique FFESs will be obtained for every arc by finding the mean of these FFESs. Finally, the FFMOTP will be transformed into a single objective Fermatean fuzzy TP (FFTP) that can be solved by applying standard algorithms. A numerical example is illustrated to support the proposed method, and the results obtained by using the proposed method are compared to those of existing techniques. Moreover, the advantages of the proposed method are also discussed.</p></abstract>

Intuitionistic fuzzy-based TOPSIS method for multi-criterion optimization problem: a novel compromise methodology

<abstract><p>The decision-making process is characterized by some doubt or hesitation due to the existence of uncertainty among some objectives or criteria. In this sense, it is quite difficult for decision maker(s) to reach the precise/exact solutions for these objectives. In this study, a novel approach based on integrating the technique for order preference by similarity to ideal solution (TOPSIS) with the intuitionistic fuzzy set (IFS), named TOPSIS-IFS, for solving a multi-criterion optimization problem (MCOP) is proposed. In this context, the TOPSIS-IFS operates with two phases to reach the best compromise solution (BCS). First, the TOPSIS approach aims to characterize the conflicting natures among objectives by reducing these objectives into only two objectives. Second, IFS is incorporated to obtain the solution model under the concept of indeterminacy degree by defining two membership functions for each objective (i.e., satisfaction degree, dissatisfaction degree). The IFS can provide an effective framework that reflects the reality contained in any decision-making process. The proposed TOPSIS-IFS approach is validated by carrying out an illustrative example. The obtained solution by the approach is superior to those existing in the literature. Also, the TOPSIS-IFS approach has been investigated through solving the multi-objective transportation problem (MOTP) as a practical problem. Furthermore, impacts of IFS parameters are analyzed based on Taguchi method to demonstrate their effects on the BCS. Finally, this integration depicts a new philosophy in the mathematical programming field due to its interesting principles.</p></abstract>

Multi-objective optimization to the transportation problem considering non-linear fuzzy membership functions

<abstract> <p>Considering the uncertainty of transporting goods from numerous origins to diverse destinations is a critical task for the decision-maker (DM). The ultimate goal of the DM is to make the right decisions that optimize the profit or loss of the organization under the vagueness of the uncontrollable effects. In this paper, mathematical models are proposed using fuzzy non-linear membership functions for the transportation problem considering the parameters' uncertainty that can help the DM to optimize the multi-objective transportation problems (MOTP) and to achieve the desired goals by choosing a confidence level of the uncertain parameters. Based on DM's selection of the confidence level, a compromise solution of the uncertain multi-objective transportation (UMOTP) is obtained along with the satisfaction level in percent for the DM. Two non-linear fuzzy membership functions are considered: the exponential and the hyperbolic functions. Using both membership functions, the sensitivity analysis was implemented by considering different confidence levels. According to the experimental results, the hyperbolic membership function gives 100% DM's satisfaction in many instances. Moreover, it shows stability against the exponential and linear functions.</p> </abstract>

Optimization of Interval Cost Multi-objective Transportation Problem in Uncertain Parameters: A Modified Least-Cost Method

The goal of this research is to focuses on multi-objective transportation problems, to reduce the cost of transporting a good from a variety of sources or origins to a range of destinations while adhering to a predetermined mathematical framework. By using interval values, the policy maker indicated the parameters of source and destination. In this work, a novel approach for finding initial basic feasible solutions (IBFS) that are extremely near to the best ones for a variety of transportation problems is presented. The proposed method can become a milestone in resolving the constraints to achieve the goal in solving transport problem. For the decision maker, regarding logistics and supply chains, it may be quite profitable. The prediction model is supported by an illustration drawn from mathematical view. All the LPPs are evaluated with the help of LINGO. Bangladesh Acad. Sci. 46(2); 155-164: December 2022

A New Approach to Solve Fully Fuzzy Multi-Objective Transportation Problem

The transportation problem is the problem of transferring goods from several sources or producers to multiple destinations or consumers in a cost-effective way, which is one of the most important problems in the supply chain management problems. The application of this problem in addition to the distribution of goods in the location and production planning problems is also important. Many real-life transportation problems encounter multiple, conflicting, and incommensurable objective functions. In addition, in real applications, due to lack of information, it is not possible to accurately estimate the parameters of this problem. Therefore, the main goal of this paper is to find the Pareto optimal solutions of fully fuzzy multi-objective transportation problem under the conditions of uncertainty. In accordingly, a new approach based on nearest interval approximation is proposed to solve the problem. Numerical examples are provided to illustrate the proposed approach and results.

Fixed-charge solid transportation problem with budget constraints based on carbon emission in neutrosophic environment

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.

Solving Multi Objective Transportation Problems using Harmonic Mean and Graph Theory

Operations research is frequently applied to economics activities, where more attention has also been directed to transportation problems (TP). Another application of this type of transportation problem can be shown as multi objective transportation problems (MOTPs). Usually, in a transportation problem, only one objective of transporting goods from source to destination is considered. However, in a MOTP, different objectives such as time, cost, and risk must be considered. The basic requirement of a multi objective transportation problem in such way as to obtain a minimum value for the objectives mentioned above. Therefore, researchers are developing various algorithms to find an efficient solution to these specific problems. Also, in this paper, we focus on presenting a new approach for an efficient solution to the MOTP. It is recommended to build the method based on the harmonic mean algorithm and spanning tree, compare its solutions with the methods of other concurrent researchers and check the effectiveness of this article. We can say that this new approach is an efficient and easy-to-understand method that provides relatively accurate solutions.