Mathematical formulation of lpp- properties and example

This ppt includes Mathematical formulation of lpp- properties and example

1. Mathematical formulation of lpp - properties and example Presented by Lakshmi p 2 nd M.com., G.F.G.C.W Holenarasipura Under the guidance of Sundar B. N. Asst. Prof. & Course Co-ordinator GFGCW, PG Studies in Commerce Holenarasipura

2. MATHEMATICAL FORMULATION OF LINEAR PROGRAMMING PROBLEM Mathematical Formulation of LPP refers to translating the real-world problem into the form of mathematical equations which could be solved. It usually requires a thorough understanding of the problem. Example: 2x+3y=66 4x+2y=44

3. Objective function: The linear programming proplems must be have a well defined objective function for optimization. For example, maximization of profit or minimization of cost or total elapsed time of the system being studied. It should be expressed as linear function of decision variables. Contraints : There are always limitation on the resources which are to be allocated among various competing activities. These resources may be production capacity, manpower, time, space or machinery. These must be capable of being expressed as linear wqulities or inequaltities in terms of decision varibles . Non-negative restriction: All the variables must assume non negative values, that is, all variables must take on values equal to or greater than zero. Therefore, the problem should not result in negative values for the variables. Properties of mathematical formulation of LPP

4. Non-negative restriction: All the variables must assume non negative values, that is, all variables must take on values equal to or greater than zero. Therefore, the problem should not result in negative values for the variables.

5. Problem on mathematical formulation of lpp A small scale industry manufactures two product A and B which are processed in a machine hour and labour hour. Product A requires 2 hours of work in a machine hour and 4 hours of work in the labour to manufacture while product B requires 3 hours of work in the machine hour and 2 hours of work in labour hour. In one day, the industry cannot use more than 16 hours of machine hour and 22 hours of labour hour. It earns a profit of 3 per unit of product A and 4 per unit of product B. Give the mathematical formulation of the problem so as to maximize profit.

6. Mathematical formulation of lpp A small scale industry manufactures two product A and B which are processed in a machine hour and labour hour. Product A requires 2 hours of work in a machine hour and 4 hours of work in the labour to manufacture while product B requires 3 hours of work in the machine hour and 2 hours of work in labour hour. In one day, the industry cannot use more than 16 hours of machine hour and 22 hours of labour hour. It earns a profit of 3 per unit of product A and 4 per unit of product B. Give the mathematical formulation of the problem so as to maximise profit. Formulation of objective function let x and y be the number of units of product A and B, which are to be produced. Here, x and y are the decision variables. Suppose Z is the profit funtion . Since one unit of product A and one unit of product B gives the profit of the 3 and 4, respectivsely , the objective funtion is maximize Z= 3x+4y The requirement and availability hours of each of the hours in manufacturing the products are tabulated as follow.

7. Mathematical formulation of lpp A small scale industry manufactures two product A and B which are processed in a machine hour and labour hour. Product A requires 2 hours of work in a machine hour and 4 hours of work in the labour to manufacture while product B requires 3 hours of work in the machine hour and 2 hours of work in labour hour. In one day, the industry cannot use more than 16 hours of machine hour and 22 hours of labour hour. It earns a profit of 3 per unit of product A and 4 per unit of product B. Give the mathematical formulation of the problem so as to maximise profit. Machine hour Labour hour Profit Product A 2 hours 4hours 3 p/u Product B 3 hours 2 hours 4 p/u Available hour p/d 16 hours 22 hours Subject to the contraints : Total hours of machine hours required for both types of product =2x+3y Total hours of labour hours required for both types of product =4x+2y hence, the constraints as per the limited available resources are: 2x+3y +≤ 16 and 4x+2y ≤ 22

8. Since the number of units produced for both A and B cannot be negative, the non negative restrictions are: thus, the mathematical formaltion of the given problem is maximize Z=3x+4y Subject to the constraints 2x+3y≤16 4x+2y≤22 And non negative restrictions x≥ 0, y≥ 0

9. Conclusion Identity the number of decision variables which govern the behavior of objective function. Identity the set of constraints on the decision variables and express them in the form of linear inequation or linear equation. Express the objective function in the form of a linear equation in the decision variable. Optimize the objective function either graphically or mathematically.

10. Reference Properties of mathematical formulation of lpp : https;// ecoursesonline,iasri.res.in /mod/resources/ view.php?id =4943 Problems on mathematical formulation of lpp : https;// youtu.be /LtpBMC6uzhw