Objective Function

An objective function represents a linear programming optimization problem and is used to solve. The equation for objective function is Z=ax+by, where Z is the value to be optimized, x and y are variables, and a and b are constants, where x > 0 and y > 0.
 

The terms related to objective function are as follows - 

• Linear Programming: The Linear programming method is used to find the optimal value of a linear objective function, it is also known as the objective function. The term linear refers to relationships among variables that form a straight line, while programming refers to the process of determining an optimal solution under the given constraint. 

• Optimization Problem: The optimization problem refers to the problems involving finding the maximum or minimum value of a linear function while meeting certain constraints, mostly represented by linear inequalities. 

• Decision variables: Decision variables are the variables that affect the outcome of the objective function. In the expression, Z = ax + by, x and y are the decision variables. It represents the quantities that can be controlled, such as resources, time, or personnel to be assigned. 

• Constraints: Constraints are the limitations on the values that the variables x and y can take in an objective function. These are usually expressed as linear inequalities in x and y. These constraints are expressed as linear inequalities involving x and y, x > 0 and y > 0. 

• Feasible Region: The feasible region is the area common to all the constraints of a linear programming problem, including non-negative constraints x ≥ 0, y ≥ 0. It includes all the possible values of x and y that satisfy the given conditions. 

• Feasible Solution: These are the points within or along the boundary feasible region, which satisfy all constraints of the objective function. To find the maximum and minimum values of the objective function, we evaluate it using a feasible solution.  

• Optimal Solution: The optimal solution, which is sometimes considered a type of feasible solution, refers to the points within the feasible region that provide the required minimum or maximum value of the objective function. 
  
   
• Optimal Value: The optimal value is the actual minimum or maximum value of the objective function at the optimal solution.      
   

The objective functions can be classified into two types, such as:

• Maximization Objective Function

• Minimization Objective Function

Maximization Objective Function: It is a mathematical expression used for maximizing the quantity in an optimization condition. It is the value obtained at one of the vertices of the feasible region formed by the constraint. 

Minimization Objective Function: It is a mathematical expression where the quantity of the function has to be minimized under optimization. The solution typically lies at one of the vertices of the feasible region. 

There are different methods of solving problems with objective functions, such as:

• Graphical Method

• Northwest Corner Method

• Iso-cost Method

Graphical Method: The graphical method involves converting the given problem to a mathematical equation and plotting it on an XY-plane. The feasible part is the region of intersection. 

 
Northwest Corner Method: First, we form an equation based on the given problem and then mark the feasible region. The objective function is then evaluated at each corner point of this region. If the feasible region is bounded, the highest and lowest values, denoted as N and n, give the maximum and minimum of the objective function. If the region is unbounded, the maximum exists only if the objective function’s open half-plane doesn’t overlap with the feasible region. Or else there is no solution

Iso-cost Method: In this method, we plot the constraints on a graph and identify the feasible region. Then plot the objective function, use the objective function, and choose any suitable value for the objective function Z, and draw its line. Draw a parallel line to the objective function line. Mark the coordinates of the corner points within the feasible region, and then decide a suitable value of Z and draw its corresponding line. 
 

In linear programming, the objective function follows two theorems that help in identifying the optimal solution:

• If R is the feasible region of a linear programming problem and Z is the objective function, then the optimal value of Z will occur at a corner point of R.

• If the feasible region R is bounded, the objective function will have both a maximum and a minimum value. 
   

The linear programming problem is solved to find to maximize or minimize an objective function, in the form Z = ax + by. In this section, we will learn how to solve an objective function. 

Step 1: Representing the objective function Z = ax + by and the constraints x ≤ k, y ≤m as a straight line on the graph. It forms a feasible region, that is, the common overlapping area. 

Step 2: Identifying the corner points of the feasible region using a graph or solving equations where the constraints' line intersects each other 

Step 3: Calculating the objective function value for each of the corner points. Then identify the maximum and minimum value.  

Step 4: The values are easy to identify if the feasible region is bounded; if it is unbounded, then:

• The maximum value of Z(M) exists only if the region does not stretch to areas where ax + by > M.

• The minimum value of Z(m) only exists if the region doesn't extend into areas where ax + by < m. 
   

The objective functions are used to quantify the goal of optimization problems and to find the optimal solution in different fields. Here are some of the real-life applications of the objective function. 

•  In supply chain optimization to find the minimum total cost while meeting demand, we use the objective function

• An objective function can be used to plan a healthy and affordable diet by determining the optimal quantity of each food item to include, based on nutritional requirements and budget constraints. 

• Objective functions are used in telecommuting, network optimization, and resource allocation to maximize bandwidth and data throughput while minimizing power consumption. 

• In business, to calculate the maximum profit, we use an objective function to find the optimal mix of products or services to maximize profit.