The fixed-charge problem deals with situations in which the economic activity incurs two types of costs: an initial "flat" fee that must be incurred to start the activity and a variable cost that is directly proportional to the level of the activity. For example, the initial tooling of a machine prior to starting production incurs a fixed setup cost regardless of how many units are manufactured. Once the setup is done, the cost of labor and material is proportional to the amount produced. Given that F is the fixed charge, e is the variable unit cost, and x is the level of production, the cost function is expressed as

The function C(x) is intractable analytically because it involves a discontinuity at x = O. The next example shows how binary variables are used to remove this intractability.

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C(x) 1o, otherwise