Proposing a Developed Gram–Schmidt Algorithm to Construct Orthogonal Unit Vectors (A Mathematical and Practical Approach)

Document Type : Original Article

Authors

1 Young Researchers and Elite Club, Arak Branch, Islamic Azad University, Arak, Iran

2 Department of Industrial Engineering, Aliabad Katoul Branch, Islamic Azad University, Aliabad Katoul, Iran.

3 School of Industrial Engineering, South Tehran Branch, Islamic Azad University, Tehran, Iran

Abstract

It is no secret that mathematics has always been a guide to solving many problems in human life. Humans, in their decision-making processes, constantly deal with various variables and indicators, and one of the approaches to analyzing these variables is the use of multi-criteria decision-making methods. However, a significant challenge in applying these techniques is the independence of criteria or indicators relative to one another. One of the modern solutions for achieving independence among variables (or indicators) is the use of the Gram-Schmidt method; however, the accuracy of this algorithm might decline when it is implemented on large-scale vectors. This paper proposes a Developed Gram–Schmidt Algorithm (DGSA). The Schmidt vectors obtained from the proposed algorithm are prone to a lower error rate than those resulting from the Gram–Schmidt algorithm. To demonstrate the superiority of the proposed algorithm, several different numerical examples have also been used. At the end of the research, a case study based on the proposed approach was also conducted at Shazand Oil Refinery (SOR) to demonstrate the applicability of this approach in a real-world example. The findings have shown that the proposed approach has relatively high accuracy.

Keywords

Main Subjects

References

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Journal of Data Analytics and Intelligent Decision-making
Volume 1, Issue 2 - Serial Number 1
September 2025
Pages 61-78

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