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During the primal simplex algorithm, a non-basic variable must be chosen to enter the basis.
Many resources on the subject choose a variable based solely on its coefficient in the row of the tableau corresponding to the objective, whereas others calculate reduced costs before making a choice.
Why is there not consistency, and is one more popular in practice?
As long as you choose something with a negative reduced cost, the simplex algorithm "works". See https://people.orie.cornell.edu/dpw/orie6300/Lectures/lec13.pdf for examples of ways you can choose the entering variable based on how much work you want to do. You can't just base it on the original cost: it has to be based on the reduced cost. Normally, this is the value in the cost row of the tableau if you are doing the full tableau method; if you are doing a more compact approach, then the reduced cost might need to be calculated (hence the value of the "first negative reduced cost" approach: you don't calculate all the reduced costs).
