Objective Function¶

Definition¶

The objective function assigns a numerical value to a decision vector. The model specifies whether this value should be maximized or minimized. In an LP it has the form

z=c_1x_1+\cdots+c_nx_n,

where the $c_j$ are given objective coefficients.

Interpretation¶

The objective expresses the criterion used to compare feasible choices. Profit maximization and cost minimization are common, but the course also considers distance, time, and flow.

A linear program written out with its parts labelled: the objective, the functional constraints and the domain restrictions; decision variables are coloured blue and parameters red.

The objective function $f(\boldsymbol{x})$ is the quantity to maximize (for example, profit) or minimize (for example, cost). Functional constraints describe what an allowed decision must satisfy, such as resource limits or demand requirements. Domain restrictions specify signs or integrality.

In the diagram, blue identifies decision variables, whose values the model determines; red identifies model parameters, the given data $c_j$ , $a_{ij}$ and $b_i$ . In general, the model is $\max/\min\ f(\boldsymbol{x})$ subject to $\boldsymbol{x}\in F$ $F$ , where the constraints define the feasible set $F$ . A solution assigns values to all variables.

Key Results¶

An objective value is a number; an optimal solution is a vector of decisions.
In a profit objective, use revenues minus relevant costs.
The direction of optimization determines what constitutes an improvement.

Examples¶

Giapetto maximizes $z=3x_s+2x_t$ $2 x_{t}$ .
The pig diet minimizes $z=0.4x_A+0.8x_B$ $0.8 x_{B}$ .

Prerequisites¶

Decision Variable

Sources¶

OR Modelling · Week 1

Symbols: element of

Last updated: 20 October 2026 03:02