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Article

Detecting All Non-Dominated Points for Multi-Objective Multi-Index Transportation Problems

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High Institute of Marketing, Commerce, and Information System, Cairo 11865, Egypt
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Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
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Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
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Department of Basic Engineering Science, Faculty of Engineering, Menofia University, Shebin El-Kom 32511, Egypt
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Department of Basic Science, Higher Technological Institute, Tenth of Ramadan City 44629, Egypt
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Author to whom correspondence should be addressed.
Academic Editor: Giovanni Leonardi
Sustainability 2021, 13(3), 1372; https://doi.org/10.3390/su13031372
Received: 3 December 2020 / Revised: 12 January 2021 / Accepted: 23 January 2021 / Published: 28 January 2021
(This article belongs to the Section Sustainable Transportation)
Multi-dimensional transportation problems denoted as multi-index are considered as the extension of classical transportation problems and are appropriate practical modeling for solving real–world problems with multiple supply, multiple demand, as well as different modes of transportation demands or delivering different kinds of commodities. This paper presents a method for detecting the complete nondominated set (efficient solutions) of multi-objective four-index transportation problems. The proposed approach implements weighted sum method to convert multi-objective four-index transportation problem into a single objective four-index transportation problem, that can then be decomposed into a set of two-index transportation sub-problems. For each two-index sub-problem, parametric analysis was investigated to determine the range of the weights values that keep the efficient solution unchanged, which enable the decision maker to detect the set of all nondominated solutions for the original multi-objective multi-index transportation problem, and also to find the stability set of the first kind for each efficient solution. Finally, an illustrative example is presented to illustrate the efficiency and robustness of the proposed approach. The results demonstrate the effectiveness and robustness for the proposed approach to detect the set of all nondominated solutions. View Full-Text
Keywords: multi-objective; multi-index transportation; parametric analysis multi-objective; multi-index transportation; parametric analysis
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MDPI and ACS Style

Abd Elazeem, A.E.M.; Mousa, A.A.A.; El-Shorbagy, M.A.; Elagan, S.K.; Abo-Elnaga, Y. Detecting All Non-Dominated Points for Multi-Objective Multi-Index Transportation Problems. Sustainability 2021, 13, 1372. https://doi.org/10.3390/su13031372

AMA Style

Abd Elazeem AEM, Mousa AAA, El-Shorbagy MA, Elagan SK, Abo-Elnaga Y. Detecting All Non-Dominated Points for Multi-Objective Multi-Index Transportation Problems. Sustainability. 2021; 13(3):1372. https://doi.org/10.3390/su13031372

Chicago/Turabian Style

Abd Elazeem, Abd Elazeem M., Abd Allah A. Mousa, Mohammed A. El-Shorbagy, Sayed K. Elagan, and Yousria Abo-Elnaga. 2021. "Detecting All Non-Dominated Points for Multi-Objective Multi-Index Transportation Problems" Sustainability 13, no. 3: 1372. https://doi.org/10.3390/su13031372

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