# Physical Interpretation of a dual of an LP

I was recently asked to physically interpret a dual of an LP for an audience who does not know mathematics/OR (without LP, dual, bounds, etc.). Though I attempted it and was very close to what the person wanted to hear (he said that), I did not meet his exact expectations (I am assuming). I am curious to know such possible interpretation(s). Please try to answer without maths and as short as possible. Also, is it possible to explain the dual without using the meaning of primal, i.e., does the dual makes any physical-sense if the primal is not present? If yes, what is it that we are minimizing in the following dual and what do the following dual constraints represent?

The optimization problem at hand:

A person has two products, A & B, with per unit cost $$C_A$$ & $$C_B$$, respectively. The person has total investment capital $$C_T$$. It is necessary to purchase at least $$A_m$$ & $$B_m$$ units of the respective products. The optimization problem is to maximize the units that can be purchased while satisfying the above requirements.

Just for reference, I formulated the following primal & dual:

Primal: Let us assume the primal variables be $$x_A$$ & $$x_B$$.

maximize $$(x_A + x_B)$$

s.t.: \begin{align} x_A &\geq A_m \tag1\\ x_B &\geq B_m \tag2\\ C_A.x_A + C_B.x_B &\leq C_T \tag3\\ A_m, B_m &\ge 0 \end{align}

Dual: Let us assume the dual variables corresponding to constraints (1), (2), (3) be $$y_A, y_B, y_T$$, respectively

minimize $$(C_T.y_T - A_m.y_A - B_m.y_B)$$

s.t.: \begin{align} C_A.y_T - y_A &\geq 1 \tag4\\ C_B.y_T - y_B &\geq 1 \tag5\\ y_A, y_B, y_T &\geq 0 \end{align}

The dual variables represent the marginal effect on the primal objective (total units purchased) per unit change in each primal constraint limit. So increasing (decreasing) the required amount $$A_m$$ of product $$A$$ by a small amount will reduce (increase) the total purchase quantity (TPQ to save me future typing) by $$y_A$$ times the change. The interpretation of $$y_B$$ is similar. Increasing (decreasing) the total capital $$C_T$$ will increase (decrease) the TPQ by $$y_T$$ times the change in capital. All these statements apply only for changes in each parameter within some range (which you can determine from the final primal solution).