Fixed charge

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A cost that is some value, say LaTeX: C, regardless of the level as long as the level is positive; otherwise the fixed charge is zero. This is represented by LaTeX: Cv, where LaTeX: v is a binary variable. If LaTeX: v=0, the fixed charge is 0; if LaTeX: v=1, the fixed charge is LaTeX: C. An example is whether to open a plant LaTeX: (v=1) or not LaTeX: (v=0). To apply this fixed charge to the non-negative variable LaTeX: x, the constraint LaTeX: x \le Mv is added to the mathematical program, where LaTeX: M is a very large value, known to exceed any feasible value of LaTeX: x. Then, if LaTeX: v=0 (e.g., not opening the plant that is needed for LaTeX: x > 0), LaTeX: x=0 is forced by the upper bound constraint. If LaTeX: v=1 (e.g., plant is open), LaTeX: x \le Mv is a redundant upper bound. Fixed charge problems are mathematical programs with fixed charges. (See Myths and Counterexamples in Mathematical Programming to avoid a misconception.)

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