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Article

The Internet Shopping Optimization Problem with Multiple Item Units (ISHOP-U): Formulation, Instances, NP-Completeness, and Evolutionary Optimization

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Faculty of Engineering “Arturo Narro Siller”, Autonomous University of Tamaulipas, Centro Universitario Tampico-Madero, Tampico 89339, Tamaulipas, Mexico
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Tecnológico Nacional de México, Instituto Tecnológico de Ciudad Madero, División de Estudios de Posgrado e Investigación, Juventino Rosas S/N, Cd. Madero C.P. 89440, Mexico
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Author to whom correspondence should be addressed.
Academic Editors: Xinchao Zhao, Xingquan Zuo, Yinan Guo and Kunpeng Kang
Mathematics 2022, 10(14), 2513; https://doi.org/10.3390/math10142513
Received: 11 June 2022 / Revised: 7 July 2022 / Accepted: 11 July 2022 / Published: 19 July 2022
In this work, we investigate the variant of the Internet Shopping Optimization Problem (ISHOP) that considers different item units. This variant is more challenging than the original problem. The original ISHOP is already known as a combinatorial NP-hard problem. In this work, we present a formal proof that the ISHOP variant considering different item units belongs to the NP-Hard complexity class. The abovementioned variant is familiar to companies and consumers who need to purchase more than one unit of a specific product to satisfy their requirements. For example, companies buy different quantities of construction materials, medical equipment, office supplies, or chemical components. We propose two new evolutionary operators (crossover and mutation) and an unfeasible solution repair method for the studied ISHOP variant. Furthermore, we produce a new benchmark of 15 synthetic instances where item prices follow a random uniform distribution. Finally, to assess our evolutionary operators, we implemented two Evolutionary Algorithms, a Genetic Algorithm (GA) and a Cellular Genetic Algorithm (CGA), and an experimental evaluation against a Water Cycle Algorithm (WCA) from the state-of-the-art. Experimental results show that our proposed GA performs well with statistical significance. View Full-Text
Keywords: internet shopping optimization problem (ISHOP); internet shopping optimization problem with units (ISHOP-U); e-commerce; evolutionary algorithms (EAs); NP-complete; NP-hard internet shopping optimization problem (ISHOP); internet shopping optimization problem with units (ISHOP-U); e-commerce; evolutionary algorithms (EAs); NP-complete; NP-hard
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MDPI and ACS Style

Ornelas, F.; Santiago, A.; Martínez, S.I.; Ponce-Flores, M.P.; Terán-Villanueva, J.D.; Balderas, F.; Rocha, J.A.C.; García, A.H.; Laria-Menchaca, J.; Treviño-Berrones, M.G. The Internet Shopping Optimization Problem with Multiple Item Units (ISHOP-U): Formulation, Instances, NP-Completeness, and Evolutionary Optimization. Mathematics 2022, 10, 2513. https://doi.org/10.3390/math10142513

AMA Style

Ornelas F, Santiago A, Martínez SI, Ponce-Flores MP, Terán-Villanueva JD, Balderas F, Rocha JAC, García AH, Laria-Menchaca J, Treviño-Berrones MG. The Internet Shopping Optimization Problem with Multiple Item Units (ISHOP-U): Formulation, Instances, NP-Completeness, and Evolutionary Optimization. Mathematics. 2022; 10(14):2513. https://doi.org/10.3390/math10142513

Chicago/Turabian Style

Ornelas, Fernando, Alejandro Santiago, Salvador Ibarra Martínez, Mirna Patricia Ponce-Flores, Jesús David Terán-Villanueva, Fausto Balderas, José Antonio Castán Rocha, Alejandro H. García, Julio Laria-Menchaca, and Mayra Guadalupe Treviño-Berrones. 2022. "The Internet Shopping Optimization Problem with Multiple Item Units (ISHOP-U): Formulation, Instances, NP-Completeness, and Evolutionary Optimization" Mathematics 10, no. 14: 2513. https://doi.org/10.3390/math10142513

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