This paper introduces a simplified presentation of a new computing procedure for solving the fuzzy Pythagorean transportation problem. To design the algorithm, we have described the Pythagorean fuzzy arithmetic and numerical conditions in three different models in Pythagorean fuzzy environment. To achieve our aim, we have first extended the initial basic feasible solution. Then an existing optimality method is used to obtain the cost of transportation. To justify the proposed method, few numerical experiments are given to show the effectiveness of the new model. Finally, some conclusion and future work are discussed.

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A Pythagorean fuzzy approach to the transportation problem

Transportation problem is the prominent class of mathematical programming problems that has a significant role in many practical transportation fields. Naturally, the transportation parameters inherently involve uncertainty in real life caused by lacking of information, imprecision in judgment, environmental factors, and etc. Therefore, it is very valuable to handle transportation problem under uncertainty aspect. The aim of this paper is to study the solution of the rough multi-objective transportation problem by supposing that the decision makers realize the transportation cost, availability and demand of the product as rough interval coefficients. The proposed approach exploits the merits of the weighted sum method to find the non-inferior solutions and it has two distinguishing features. Firstly, the proposed approach characterizes the surely Pareto optimal solution through converting the lower interval into two crisp transportation problems. Secondly, the proposed approach characterizes the possibly Pareto optimal solution through decomposing the upper interval into two crisp transportation problems. Furthermore, the expected nondominated value is applied to obtain the optimal compromise solutions of multi-objective transportation problem in rough environment. The presented approach is showed with rough multi-objective optimization problem as numerical illustration, where a wide set of the expected compromise solution ranged from 15.75 to 25.8 can be obtained. Furthermore, the investigation on the rough multi-objective transportation problem is conducted a real thought-provoking case study, where the optimal rough interval of transportation cost ranged from 97 to 314 can be achieved. With the adoption of rough environment modeling, a wide variate of optimal solutions can be achieved that can help the decision maker to extract the best compromise alternative according to practical situations. This represents a novel contribution to the decision making field and profit satisfaction models.

A novel approach for solving rough multi-objective transportation problem: development and prospects

In this study criterion of maximum profit intensity for transportation problems, in contrast to the known criteria of minimum expenses or minimum time for transportation, is considered. This criterion synthesizes financial and time factors and has real economic sense. According to the purpose of this paper, the algorithm of the solution of such a transportation problem is constructed. It is shown that the choice is carried out among Pareto-optimal options. Moreover, the factor of time becomes defining for the high income from transportation, and the factor of expenses – at low ones. Not absolute but relative changes of numerator and denominator become important when the criterion represents the fraction (in this case – the profit intensity as the ratio of profit to time). A nonlinear generalization of such transportation problem is proposed and the scheme of its solution in a nonlinear case is outlined. Graphic illustrations of Pareto-optimal and optimal solutions of transportation problem by profit intensity criterion are also given.

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Profit intensity criterion for transportation problems

A systematic and organized overview of various existing transportation problems and their extensions developed by different researchers is offered in the review article. The article has gone through different research papers and books available in Google scholar, Sciencedirect, Z-library Asia, Springer.com, Research-gate, shodhganga, and many other E-learning platforms. The main purpose of the review paper is to recapitulate the existing form of various types of transportation problems and their systematic developments for the guidance of future researchers to help them classify the varieties of problems to be solved and select the criteria to be optimized.

A Comprehensive Literature Review on Transportation Problems

Companies have been trying continuously to reduce their logistics costs in the current competitive markets. Warehouses are important components of the logistics systems and they must be managed effectively and efficiently to reduce the production cost as well as maintain customer satisfaction. Order-picking is the core of warehouse operations and an order-picking system (OPS) is essential to meet customer needs and orders. Failure to perform the OPS process properly results in high costs and customer dissatisfaction. This research aims to investigate the state of the art in the adoption of OPS and provide a broad systemic analysis on main operating strategies such as simultaneous consideration of order assignment, batching, sequencing, tardiness, and routing need. This study reviews 92 articles, classifies combinations of tactical and operational OPS problems, and provides guidelines on how warehouse managers can benefit from combining planning problems, in order to design efficient OPS and improve customer service. Combining multiple order-picking planning problems results in substantial efficiency benefits, which are required to face new market developments.

A Survey of the Literature on Order-Picking Systems by Combining Planning Problems

An innovative exact method for solving fully interval integer transportation problems

An innovative method, namely, level maintain method, is proposed for finding all efficient solutions to a bottleneck-rough cost interval integer transportation problem in which the unit transportation cost, supply, and demand parameters are rough interval integers and the transportation time parameter is an interval integer. The solving procedure of the suggested method is expressed and explained with a numerical example. The level maintain method will dispense the necessary determined support to decision-makers when they are handling time-related logistic problems in rough nature.

On Bottleneck-Rough Cost Interval Integer Transportation Problems

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G. Natarajan

Papers

An innovative exact method for solving fully interval integer transportation problems

On Bottleneck-Rough Cost Interval Integer Transportation Problems

Algorithmic Approach for Solving Transportation Problems