Anatomy of an LP: Decision Variables, Objective, Constraints

Identify and formulate the three structural pieces of a linear program: decision variables, the objective function, and constraints (including non-negativity).

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Tutorial

The Three Pieces of a Linear Program

Every linear program (LP) is built from three pieces:

Decision variables — the unknown quantities under our control.
Objective function — a linear function of the decision variables we want to maximize or minimize.
Constraints — linear inequalities or equalities the variables must satisfy.

This tutorial focuses on the first piece.

Decision variables represent quantities the decision-maker chooses. Typical examples are units to produce, dollars to invest, hours to allocate, or kilograms to ship. We give them symbols like $x_1, x_2, \ldots$ and we always state what each one measures, including units and time frame.

For example, suppose a farmer must decide how to split land between wheat and corn for the upcoming season. Two natural decision variables are

x_1 = \text{acres of wheat planted this season},

x_2 = \text{acres of corn planted this season}.

Notice that the definition pins down what is being chosen and in what units. Without the units and time frame, the rest of the LP would be ambiguous.