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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?

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  • $\begingroup$ 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)? $\endgroup$
    – prubin
    Feb 28 '20 at 18:46
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    $\begingroup$ 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. $\endgroup$
    – A.Omidi
    Feb 29 '20 at 5:36
  • $\begingroup$ @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. $\endgroup$
    – Sakazuki Akainu
    Mar 10 '20 at 4:44
  • $\begingroup$ @A.Omidi that link was super helpful! Thank you! $\endgroup$
    – Sakazuki Akainu
    Mar 10 '20 at 4:45

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