Profile
Settings
Refer your friends
Sign out
A furniture dealer sells two items–tables and chairs. He has Rs.30000 to invest. He also has a storage space of at most 40 pieces. A table costs Rs.2000, and a chair costs Rs.500.He also estimates that selling one table makes a profit of Rs.200 and selling one chair makes a profit of Rs.50. He wants to know how many tables and chairs he should buy to maximize his profit by assuming that he can sell all the items he buys.
Such kinds of problems in which we need to find maximum/minimum profit from a general class of problems known as optimization problems. One of the most important problem types in optimization problems is linear programming.
TERMS IN LINEAR PROGRAMMING PROBLEMS
The main part of the linear programming problems is finding the optimum values of a linear, but the variables provided must be non-negative and satisfy all the inequalities. Thus the used mathematical relations are linear relations.
Let us take a simple relation
Z=ax+by
Where the relation Z is a linear objective function.
a and b are known as decision variables.
The restriction on the variables in a linear programming problem is called constraints.
GRAPHICAL METHOD OF SOLVING LINEAR PROGRAMMING PROBLEMS
The common region that is determined by all constraints, including the non-negative constraints x, y ≥ 0 in a linear programming problem, is known as a feasible region.
The region other than feasible regions is known as non-feasible regions.
Points within the feasible region are known as feasible points or feasible solutions.
The points located in a region outside of the feasible region are known as non–feasible points or non-feasible solutions.
Any point that gives the maximum and minimum value of a function is known as an optimal solution or feasible solution.
THEOREMS FOR OPTIMIZATION PROBLEMS
THEOREM 1
Let us consider R as a feasible region and the Z=ax+by be the objective function. The objective function Z has optimal values. The variables x and y are given in the limited constraints. So this optimal value must occur in the corner point of the feasible region.
THEOREM 2
Let us consider R as a feasible region and the Z=ax+by be the objective function. The objective function Z has optimal values. If R is bounded then the maximum and minimum value of z must lie on R and each of these occur at corner points of R.
Suppose if R is an unbounded region then the optimal values ( maximum and minimum ) may not occur in R.
CORNER POINT METHOD
The method of solving linear programming problems is known as the corner point method.
Corner point method in solving a problem of linear programming:
The method comprises of the following steps:
- Find the feasible region of the linear programming problem and determine its corner points (vertices).
- Evaluate the objective function Z = ax + by at each corner point. Let us assume that M and m be the largest number and smallest number respectively.
- If the feasible region is bounded, M and m respectively are the maximum and minimum values of the objective function.
If the feasible region is unbounded, then
- M is the maximum value of the objective function if the open half-plane determined by ax + by > M has no point in common with the feasible region.
- m is the minimum value of the objective function, if the open half-plane determined by ax + by < m has no point in common with the feasible region.
DIFFERENT TYPES OF LINEAR PROGRAMMING PROBLEMS
Manufacturing problems: The number of units of different products which should be produced and sold by a firm when each product requires fixed manpower, machine hours, labour hours per unit of product, warehouse spaces per unit of the outputs, etc., in order to make maximum profit.
Diet problems: In this problem, the number of different nutrients to be included in the diet is found to minimize the cost of nutrient materials such that it contains the minimum cost of each nutrient.
Transport problems: In these problems, we determine a schedule for transportation to find the cheapest way of transport of different materials situated at different locations to different markets.
Marketing Management problems: It helps marketers analyse their business’s constraints. It also provides the salesperson with the shortest distance to reach their destinations. A logistic head can find the optimal distribution schedule for transporting the product from different places to various market locations to minimize the transport cost for sending those goods.
Financial Management problems: Linear programming helps the financial firms, mutual fund firms, and banks select the investment portfolio of shares, bonds, etc., to make the return on investments maximum.
Agricultural management: Linear programming problems also help in managing agricultural business by finding good investments in seasonal crops and finding the amount of crops to be produced to earn profits.
Conclusion
We can conclude that linear programming is a mathematical method for determining the most efficient use of resources and their optimization. Managers use the process to help them make the best use of limited resources like money, time, materials, and machinery.
