Canonical form and Standard form of LPP

This ppt includes Canonical form and Standard form of LPP

1. S Subject:- Topic::- Canonical form and Standard form of LPP Under the guidance of Sundar B. N. Asst. Prof. & Course Co-ordinator GFGCW, PG Studies in Commerce Holenarasipura

2. GGa Presented By, GAGANA N. R. VI Semester M.Com Govt. First Grade College for WOMEN, PG Studies Center in COMMERCE, Holenarasipura . UNIVERSITY OF MYSORE G.F.G.C for WOMEN, PG STUDIES IN COMMERCE

3. Introduction of LPP Canonical form of LPP Standard form of LPP Difference Between Canonical form and Standard form of LPP Conclusion Reference Content:

4. Introduction: The hand account is an indispensable necessities of life. It is used to record what is happening in your daily life. The hand account is an indispensable necessities of life. It is used to record what is happening in your daily life. LPP: Linear programming is a method to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special Case of mathematical programming. The hand account is an indispensable necessities of life. It is used to record what is happening in your daily life. . LPP

5. For Problems:: * First we formulate the problem LPP next. Solve it * The objective in a LP is to maximize profit or minimize cost, as the may be, subject to a number of limitations know as constraints. * It is a standard procedure, that is standard form and conic LPP Gg

6. Canonical Form:- Canonical form of LPP: Ca nonical form of standard LPP is a set of equations consisting of the 'objective function' and all the 'equality constraints' (standard form of LPP) ex pressed in canonical form. Understanding the canonical form of LPP is necessary for studying simplex method, the most popular method of solving LPP. LPP

7. Characteristics of canonical form: Characteristics of canonical form 1 . Objective function should be of maximisation from. If it is given in minimisation from, it should be converted into maximisation form. 2 . All the constraints should be of " ≤ type, except for non-negative restrictions. Inequality of " ≥ t ype if any , should be changed to on inequality of the " ≤ type (by multiplying both the side of - 1) 3 . All variables should be non - negative - difference of tow non-negative variables 4. Canonical form of a set of linear equation will be discussed next. 5. Canonical from of LPP is essential for simplex method. 6. This is know as canonical form of LPP. LPP

8. How to convert general problem to canonical form? Ex ample (problem) : Minimize the Z = 2x + y + 6z Subject to constraints : 4x + 3y + z ≤ 5 2x - y + 3z ≤ 7 x +5y ≤ 3 X, Y . ≥ an d z is unrestricted in sign LPP

9. ≤ Solution: Step 1: Objective function (of) Minimize z =2x + y +6z Now we have to minimize - z = -2x - y - 6z Maximize the z'= -2x - y - 6z Where z'= -z Step 2: consider > constraints So now consider 1st constraint (Multiplied by - 1) 4x + 3y + z > 5 (x-1) =-4x - 3y - z < -5 Minimize the Z = 2x + y + 6z Subject to constraints : 4x + 3y + z ≤ 5 2x - y + 3z ≤ 7 x +5y≤ 3 X, Y > 0 and z is unrestricted in sign Note: In this presentation we consider, < is . ≤ > is. ≥

10. Minimize the Z = 2x + y + 6z Subject to constraints : 4x + 3y + z < 5 2x - y + 3z <7 x +5y < 3 X, Y > 0 and z is unrestricted in sign St ep3: Final mathematical formulation Maximize the z'= -2x - y - 6z Subject too Constraint -4x - 3y - (z-z) I < -5 2x-y +3(z-z) < 7 x + 5y < 3 Non-negative restriction x > 0, y > 0, z > 0, z > 0 LPP

11. Standard Form of LPP:- T he simple methods, which is the procedure we will use for solving linear program's that are in a fixed format we will call the standard form. Standard form meaning is maths is defined as representation or notation of that particular element . It depends on the subject weather it is numbers, or equation or a line .

12. It is developing general procedure pr oblem. It is standard procedure. It is simplexed procedure of canonical form I t is a simplified version of canonical form Picture LPP

13. The hand account is an indispensable necessities of life. It is used to record what is happening in your daily life. nsable necessities of life. It is used to record what is happening in your daily life. The hand account is an indispensable necessities of life. It is used to record what is happening in your daily life. Characteristics of Standard Form:- 1 - The objective function should be of maximisation form. 2 - The right side element of each constraint should be non-negative (if not X-1) 3 - All constrains should be expressed in the form of equations (equalities), I expect for the non-negative restrictions by augmenting slack or s ur plus variables !). For inequalities of the of the "<" I type, we add the slack variables ||)If the inequalities are of the ">" type, we subtract surplus variables LPP

14. How to convert to standard Form of LPP? Example (problem) Minimize z = 2x + y + 4z Subject to constraints : -2x + 4y < 4 X + 2y + z > 5 2x + 3z < -2 x > 0, y > 0, z > 0 2x + 3z + s3 = 2 Non - negatives X > 0, Y > 0, Z > 0, S1> 0, S2 >0, S3 > 0

15. Example (problem) Minimize z = 2x + y + 4z Subject to constraints : -2x + 4y < 4 X + 2y + z > 5 2x + 3z < -2 x > 0, y > 0, z > 0 2x + 3z + s3 = 2 Non - negatives X > 0, Y > 0, Z > 0, S1> 0, S2 >0, S3 > 0 Step 1 : • Consider objective function Minimize the Z = 2x + y + 4z We have to maximize the of Z'= -2x - y - 4z Where Z' = -Z Step 2 :• consider the negative right constraints 2x + 3z < -2 The right side of the constraints should be non-negative : so ( multiplied by -1) -2x - 3z < 2

16. Step 3: Inequalities are to be converted T o equations -2x + y + s1 =4 X + 2y + z - s2 =5 2x + 2z + s3 =2 Step 4 : Standard form of LPP maximize Z'= -2x - y - 4z + s1 - s2 +s3 Subject to the constraints - 2x + 4y + s1 =4 X +2y + z - s2 =5 2x + 3z + s3 = 2 Non - negatives X > 0, Y > 0, Z > 0, S1> 0, S2 >0, S3 > 0 Minimize z = 2x + y + 4z Subject to constraints : -2x + 4y < 4 X + 2y + z > 5 2x + 3z < -2 x > 0, y > 0, z > 0 2x + 3z + s3 = 2 Non - negatives X > 0, Y > 0, Z > 0, S1> 0, S2 >0, S3 > 0 LPP

17. Standard form : 1. S andard form is a simplified version of canonical form that represents boolean outputs of digital circuit's using boolean Alge bra. 2 . Simple Canonical form : 1.can onical from is a way of representing boolean outputs of digital circuits using Boolean algebra 2. More complex Difference between canonical form and standard form :•

18. Standard form and canonical form of LPP: * A given linear program can be put in different equivalent form by suitable Manipulation. T wo forms in particular will be useful. These are the standard and canonical form, A linear program is said to be in standard format is all restrictions are e qualities and all variables are non negative. The simplex Method is designed to be applied only after the problem is putting in standard form

19. * A minimisation problem is in canonical form is all variables are non negative and all constraints are of the type. A maximisation problem is in canonical form is all variables are non negative and all constraints are of the type. A maximisation problem is IN canonical form is all variables are none negative And all constraints ate of the type.

20. Conclusion : Linear programming is very important Mathematical technique which enables Manager to arrive at proper decisions regarding his area of work. Thus it is very important part of operation research . The main objective is to maximisation of profit and minimisation of cost. It is very useful for optimization solution. LPP

21. Reference:- https://www.matem.unam.mx/~omar/math340/std-form.html https://youtu.be/PJG8zj5YkpI https://youtu.be/fvTHjfxKgz0 https://www.slideshare.net/shamjithkeyem/numerical-analysis-canonica l https://pediaa.com/what-is-the-difference-between-canonical-and-standard-form/