Inventory balance equation

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The constraint that says that the amount of inventory in the next time period must equal the current inventory plus what is produced or purchased minus what is lost or sold. Let LaTeX: y(t) be the inventory at the beginning of period LaTeX: t, with LaTeX: y(0) given. Then, the inventory equation is:

LaTeX: 
y(t+1) = ay(t) + P(t) - S(t),

where LaTeX: P(t) is the level of production (or somehow acquired), and LaTeX: S(t) is the level of sales (or somehow consumed). Typically, LaTeX: a=1, but if LaTeX: a < 1, it is called a loss factor, and if LaTeX: a > 1, it is called a gain factor.

The language used is for the inventory control in the production scheduling problem, but this has become a standard system of equations that appears in many mathematical programs. Thus, the meaning of the variables can be substantially different. One example is the

steel beam assortment problem.

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