1. Meaning of an Objective Function
The objective function is the main goal of a linear programming problem written as a mathematical expression. It tells what needs to be maximized or minimized.
Usually the objective is related to profit, cost, time, output, or some quantity that needs to be improved.
It is always a linear expression in the decision variables, such as:
\( Z = c_1 x + c_2 y + \dots \)
Here, \( x, y, \dots \) are decision variables and \( c_1, c_2, \dots \) are constants (like profit or cost per unit).
2. Types of Objective Functions
There are mainly two types of objective functions in linear programming.
2.1. Maximization Objective
Used when the aim is to make something as large as possible, such as profit, output, or efficiency.
Typical form:
\( \text{Maximize } Z = c_1 x + c_2 y \)
Example: maximize total profit from two products.
2.2. Minimization Objective
Used when the aim is to make something as small as possible, such as cost, time, or waste.
Typical form:
\( \text{Minimize } Z = c_1 x + c_2 y \)
Example: minimize total transportation cost or total labour cost.
3. Writing an Objective Function from a Verbal Statement
To form the objective function, first identify what is being optimized and then relate it to the decision variables.
Steps:
- Choose decision variables (for example, number of units of each product).
- Find the contribution of each variable to the goal (profit per unit, cost per unit, etc.).
- Multiply contribution by the variable and add them to build a linear expression.
3.1. Example: Maximizing Profit
Suppose a workshop makes two products, A and B.
- Let \( x \) = number of units of A
- Let \( y \) = number of units of B
- Profit per unit of A = 40
- Profit per unit of B = 60
Total profit:
\( Z = 40x + 60y \)
Objective function: Maximize \( Z = 40x + 60y \).
3.2. Example: Minimizing Cost
Suppose there are two transportation routes.
- Let \( x \) = number of units sent on route 1
- Let \( y \) = number of units sent on route 2
- Cost per unit on route 1 = 8
- Cost per unit on route 2 = 10
Total cost:
\( Z = 8x + 10y \)
Objective function: Minimize \( Z = 8x + 10y \).