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Objective function is prominently used to represent and solve the optimization problems of linear programming. The objective function is of the form Z = ax + by, where x, y are the decision variables. The function Z = ax + by is to be maximized or minimized to find the optimal solution. Here the objective function is governed by the constraints x > 0, y > 0. The optimization problems which needs to maximize the profit, minimize the cost, or minimize the use of resources, makes use of an objective function.
The objective function is used across industry, commerce, management, applied sciences to solve numerous real-life problems. Let us learn more about solving the objective function, its theorems, applications, with the help of examples, FAQs.
What Is Objective Function?
The objective function is needed to solve the optimization problems. A linear representation of the form Z = ax + by, where a, b are constraints, and x, y are variables, which have to be maximized or minimized is called an objective function. The variables x and y are called the decision variables. An objective function is governed by a few constraints, some of which are x > 0, y > 0.
Objective Function: Z = ax + by
Objective function of a linear programming problem is needed to find the optimal solution: maximize the profit, minimize the cost, or to minimize the use of resources, right deployment of resources. Objective function in LPP has wide application in representing problems of commerce, industry, and applied sciences.
Terms Related To Objective Function
The following terms help in easily understanding the concept of the objective function
- Linear Programming: The concept of linear programming deals with finding the optimal value of a linear function, which has been referred to as an objective function. The term linear refers to all the mathematical relationships used in the problem are linear relationships, and the term program refers to the method of determining a particular program or plan of action.
- Optimization Problem: A problem that seeks to maximize or minimize a linear function subject to certain constraints as determined by a set of linear inequalities is called an optimization problem.
- Decision Variables: The resources to be utilized, the time to be allocated, or the people to be deployed can all be the decision variables for an objective function. The objection function z = ax + by has the variables x, and y as the decision variables.
- Constraints: The limitations of the values which the variables x, y can take are the constraints of the objective function. The constraints of the objective function are expressions as linear inequalities in x and y. The constraints x > 0, y > 0 are the simplest representation of non negative restrictions.. As an example, the number of boys is represented as x, and the number of girls is represented as y, and the constraint exists that the number of children who can be accommodated in the bus is less than 60. This constraint can be expressed as an inequality x + y < 60.
- Feasible Region: The common region determined by all the constraints including non-negative constraints x > 0, y > 0 of a linear programming problem is called the feasible region.
- Feasible Solution: These are the points in the feasible region or on the boundary of the feasible region, representing the feasible solution of the objective function. The feasible solution helps to find the maximum and minimum values of the objective function.
- Optimal Solution: The optimal solution is also sometimes referred to as a feasible solution. The points in the feasible region in the graph of the objective function, which gives the required minimum or maximum value for the objective function is referred as the optimal solution.
- Optimal Value: Optimal Value is the value at which the solution of the objective function has a maximum value or a minimum value. This in real-life value could be the minimum cost, the maximum profit, or the optimal use of resources.
Theorems Of Objective Function
The two important theorems of the objective function of a linear programming problem are as follows.
- Theorem 1: Let there exist R the feasible region (convex polygon) for a linear programming problem and let Z = ax + by be the objective function. When Z has an optimal value (maximum or minimum), where the variables x and y are subject to constraints described by linear inequalities, this optimal value must occur at a corner point* (vertex) of the feasible region.
- Theorem 2: Let R be the feasible region for a linear programming problem, and let Z = ax + by be the objective function. If R is bounded**, then the objective function Z has both a maximum and a minimum value on R and each of these occurs at a corner point (vertex) of R.
Solving An Objective Function
The feasible region of the objective function of a linear programming problem is obtained by forming a graph of the objective function, finding the corner points, and then finding the maximum and minimum values. Let us try to understand this in the below steps.
- The objective function z = ax + by is represented as a linear equation of a line, and the constraints x < k, and y < m are also represented as lines in the graph.
- Identify the corner points by simple inspection and by finding the point of intersection of the function Z, constraint lines, and the axis lines.
- Further, evaluate the objective function Z for each of the corner points, and identify the maximum value M and minimum value m, for the objective function.
- If the feasible region is bounded it is easy to identify M, m, and if the feasible region is not bounded, we can check the below two steps to identify the maximum and minimum value.
- M is the maximum value of Z, if the open half-plane determined by ax + by > M has no point in common with the feasible region. Otherwise, Z has no maximum value.
- Similarly, m is the minimum value of Z, if the open half-plane determined by ax + by < m has no point in common with the feasible region. Otherwise, Z has no minimum value.
Application Of Objective Function
The objective function has great application in functions that are used for optimizing the profits, minimizing the cost, and making optimal use of limited resources. The various applications of the objective function are as follows.
- Manufacturing Problem: The number of units of different products to be produced by a manufacturing firm is dependent on the machine time, number of manhours, warehouse space per unit item. All of these needs to be considered when we are solving the problem of minimizing the cost and allocating the needed resources.
- Diet Problems: The .proportion in which the different constituents of the diet are to be included in the food so as to optimize the nutrients received, and also to reduce the cost of the diet is a linear programming problem that can be solved using an objective function.
- Transportation Problem: The different routes which have to be followed so as to reduce the distance, increase the number of dropping points, reduce the fuel cost, is an optimization problem, which can be presented as an objective function and solved for the best solution.
Examples of Objective Function
Example 1: A furniture dealer has to buy chairs and tables and he has total available money of $50,000 for investment. The cost of a table is $2500, and the cost of a chair is $500. He has storage space for only 60 pieces, and he can make a profit of $300 on a table and $100 on a chair.
Express this as an objective function and also find the constraints.
Let us consider the number of tables as x and the number of chairs as y. The cost of a table is $2500, and the cost of a chair is $500, and the total cost cannot exceed more than $50,000.
Constraint - I: 2500x + 500y < 50000 OR 5x + y < 100.
The dealer does not have storage space for more than 60 pieces. This can be represented as a second constraint.
Constraint - II: x + y < 60
There is a profit of $300 on the table and $100 on the chair. The aim is to optimize the profits and this can be represented as the objective function.
Objective Function: Z = 300x + 100y.
Therefore, the constraints are 5x + y < 100, x + y < 60, and the objective function is Z = 300x + 100y.
Example 2: A manufacturing company makes two kinds of instruments. The first instrument requires 9 hours of fabrication time and one hour of labor time for finishing. The second model requires 12 hours for fabricating and 2 hours of labor time for finishing. The total time available for fabricating and finishing is 180 hours and 30 hours respectively. The company makes a profit of $800 on the first instrument and $1200 on the second instrument.
Express this linear programming problem as an objective function and also find the constraint involved.
Let the two kinds of instruments be such that there are x number of the first instrument and y number of the second instrument. Given that 9 hour of fabrication time and 1 hour of finishing time is needed for each of the x number of the first instrument. Also 12 hours of fabricating time and 2 hours of finishing time is required for y number of the second instrument. Further, there are a total of 180 hours for fabricating and 30 hours for finishing. These can be defined as the two constraints.
Constraint - 1(For Finishing): 9x + 12y < 180 OR 3x + 4y < 60
Constraint - II(For Fabricating): x + 2y < 30.
The company makes a profit of $800 on each of the x numbers of the first instrument, and a profit of $1200 on each of the y numbers of the second instrument. The aim is to maximize the profits and this can be represented as an objective function.
Objective Function: Z = 800x + 1200y
Therefore the two constraints are 3x + 4y < 60, x + 3y < 30, and the objective function is Z = 800x + 1200y.
Practice Questions on Objective Function
FAQs on Objective Function
What Is Objective Function?
An objective function is a linear equation of the form Z = ax + by, and is used to represent and solve optimization problems in linear programming. Here x and y are called the decision variables, and this objective function is governed by the constraints such as x > 0, y > 0. The objective function is used to solve problems that need to maximize profit, minimize cost, and minimize the use of available resources.
How To Solve An Objective Function?
The objective function Z = ax + by is generally solved graphically. The graph of an objective function is a line and the constraints and the axis lines in the coordinate plane together form the feasible region. The corner points are identified and the same is substituted in the objective function to find the maximum value M and the minimum value m, of the objective function.
Why Do We Call It An Objective Function?
The objective function is referred to by this name since it is used to find the optimal solution of a linear programming problem. The objective function is solved to find the maximum or minimum value, by graphing functions in the coordinate system.
Where Do We Use The Objective Function?
The objective function is used in problems where the decision variables have constraints. The problem of minimizing the cost, maximizing the profits, within the limited resources is easily solved by representing them as linear equations. The objective function is widely used in commerce and industry, management, applied sciences.
Write A Few Examples Of Objective Function?
The objective function is prominently used in manufacturing problems, diet problems, logistic problems. The limitation of machine time, limited man hours, warehouse space, are the constraints that are applied to an objective function to solve the manufacturing problem. The right proportion of diet with the required number of nutrients and at a low cost is to be solved as an objective function. For solving the logistics problem, we need to consider the right route, to reduce the fuel cost and time, and find the optimal solution of the objective function.