# Derivations for two formulae for obtaining optimal dual variable values from the optimal primal tableau

We're being taught Industrial Engineering and Operations Research for the first time this semester. Referring to the book by Hamdy A. Taha, I noticed the mention of two formulae for swiftly obtaining the dual variable optimal solutions from the primal optimal tableau:

Method 1. $${\text{Optimal value of}\\dual\text{ variable }y_i}\quad=\quad{{\text{Optimal primal }z\text{-coefficient of }\\starting\text{ basic variable }x_i}\quad+\quad{Original\text{ objective}\\\text{coefficient of }x_i}}$$ Method 2. $${\text{Optimal values}\\\text{of }dual\text{ variables}}\quad=\quad{\text{Row vector of }original\\\text{objective coefficients of}\\\text{optimal }primal\text{ basic}\\\text{variables}}\quad\times\quad{\text{Optimal }primal\text{ inverse}}$$

Does anyone have an intuitive explanation from how these are derived?

• Welcome to OR SE. By "intuitive" do you mean an outline of the logic behind a derivation, or do you mean something expressed in terms of "costs" (or "profits") and marginal costs or profits (i.e., something in economic terms)? – prubin Feb 28 at 18:46
• To calculate what you looking for Based on the simplex method (algebra expressions), you can use $z_j - c_j = c_bB^{-1}a_j - c_j$ to figure out objective function row in each iterate in the tableau. Also, this link would be useful. – A.Omidi Feb 29 at 5:36
• @prubin no sorry, we're not learning this strictly from an econ perspective. I was looking for the logic behind the derivation. Thanks, and sorry for the late response! Just in case, I'm from a Mechanical Engineering undergraduate here. – Sakazuki Akainu Mar 10 at 4:44
• @A.Omidi that link was super helpful! Thank you! – Sakazuki Akainu Mar 10 at 4:45