Steel beam assortment problem

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A steel corporation manufactures structured beams of a standard length, but a variety of strengths. There is a known demand of each type of strength, but a stronger one may fulfill demand (or part thereof) for another beam (but not conversely). The manufacture of each type of steel beam involves a fixed charge for its setup. In addition, there is a shipping cost proportional to the difference in the demanded strength and the actual strength, and proportional to the quantity shipped.

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".)

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