 # Steel beam assortment problem

Let $LaTeX: N$ = number of varieties of strengths $LaTeX: D(t)$ = demand for beam of strength $LaTeX: s_t$ (where $LaTeX: \textstyle s_1 \ge s_2 \ge \dots \ge s_N$ ) $LaTeX: x(t) =$ amount of beams of strength $LaTeX: s_t$ manufactured $LaTeX: \textstyle p_t(x(t)) =$ manufacturing cost of $LaTeX: x(t)$ units of beam of strength $LaTeX: s_t$ (incl. fixed charge) $LaTeX: y(t) =$ total excess of beams of strength $LaTeX: \textstyle s_1, \dots , s_t$ before fulfilling demand $LaTeX: \textstyle D(t+1), \dots , D(N) h_t(y(t)) =$ shipping cost $LaTeX: \textstyle ( = c [ s_{(t+1)}- s_t] \min \{y(t), D(t)\}).$
Although $LaTeX: t$ does not index time, the mathematical program for this problem is the same form as the production scheduling problem, using the inventory balance equations to relate $LaTeX: y$ and $LaTeX: x$. This is valid because $LaTeX: \textstyle s_1 \ge s_2 \ge \dots \ge s_N$ implies $LaTeX: y(t)$ can be used to fulfill demand $LaTeX: \textstyle D(t+1) + D(t+2) + \dots + D(N).$ (Also, note that here $LaTeX: h_t$ is not a "holding cost".)