On inequality constraints in well-known optimisation problems: properties and applications

Abstract

This work studies transportation, assignment, transshipment, shortest path, and dynamic lot sizing problems. Particularly, it shows how to modify their formulations to handle excess supply and node elimination. The paper proceeds by using the conservation of flow to derive the original constraints of these problems. Thereafter, a general form of the conservation of flow is used to reformulate these problems. The results are problems with inequality constraints. This paper shows that solving each of these new formulations returns the same optimal solution as its original form. It also shows how problems with inequality constraints can handle excess supply and node elimination. Along the way, some additional properties of these alternative formulations are explored.

Keywords:

• Optimization problems
• constraints
• conservation of flow

Disclosure statement

The author declares that there is no conflict of interest.

Data availability statement

This is a theoretical paper and deals with proofs. There is no particular data except for the formulations, statements, proofs, and applications that are written in the paper.

Additional information

Funding

This work was partially supported by the US National Science Foundation under grant number 1719514.

Notes on contributors

H. Farhangi

Hadi Farhangi holds a Ph.D. in systems engineering from the Missouri University of Science and Technology and a master's degree in socio-economic systems engineering from the Sharif University of Technology. He has been a faculty at Savannah State University since 2017. His research is in Logistics, Supply Chain Management, and Systems Architecting. His teaching is related to Operations Research, Operations Management, Supply Chain Management, and Quantitative Studies.