Related Terminology: Constraints, Objective Function and Optimization

Decision variables. The first step in setting up any LPP is to name the unknown quantities that are actually under the decision-maker's control -- for instance, the number of units of two products to manufacture, or the number of units of two foods to buy. These are written as $x$ and $y$ (or any other symbols), and every other quantity in the problem -- profit, cost, resource usage -- is expressed in terms of them.

Objective function. The quantity to be optimized (made as large or as small as possible) is written as a linear function of the decision variables,

where $a,b$ are given constants (e.g. profit per unit, or cost per unit). The problem statement always specifies whether $Z$ is to be maximized (e.g. profit, output, reach) or minimized (e.g. cost, distance, time, wastage) -- this choice is called the optimization to be performed, and it changes which corner point of the feasible region turns out to be optimal (Section 6).

Constraints. Every real resource is available only in a limited amount, and this is expressed as a linear inequality in the decision variables, called a constraint. If, for example, one unit of a product needs $p$ hours of a machine that is available for at most $H$ hours, and a second product needs $q$ hours of the same machine, then producing $x$ units of the first and $y$ of the second uses $px+qy$ hours of the machine, and the resource limit is written

A minimum requirement (e.g. a diet must supply at least a certain amount of a nutrient, or a delivery target must be met) is written the other way, with $\ge$ instead of $\le$. A constraint is called non-trivial when it genuinely restricts the decision variables beyond simply requiring them to be non-negative; this chapter's syllabus caps the number of non-trivial constraints at three, precisely so that the resulting feasible region -- typically a triangle, quadrilateral, or (with three non-trivial constraints) occasionally a pentagon -- can be drawn and its corners read off accurately by hand.

Non-negative restrictions. Because a decision variable in a real LPP almost always represents a physical count or quantity (units produced, kilograms bought, vehicles used), it is never allowed to be negative. This is written as the pair of constraints $x\ge0,\ y\ge0$, and is included in essentially every LPP even though it is not usually called out as one of the "non-trivial" constraints. …