PPT-Unit-03-MB0048

Program Semester Subject Code Subject Name Unit number

: MBA : II : MB0048 :Operation Research :3

Unit TitleLecture Number Lecture Title

:Simplex Method: 4 &5 :Introduction to simplex method

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Unit-3 Simplex method

Objectives : Create a standard form of LPP from the given hypothesis Apply the simplex algorithm to the system of equations Interpret the big M-technique Discuss the importance of the two phase method

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Lecture Outline

Overview of simplex method Standard form of LPP Fundamental theorem of LPP Simplex Algorithm Penalty cost/ Big-M method Two-phase method



Overview of Simplex Method

Efficient technique which can be applied for solving LPP of any magnitude involving two or more decision variables Objective function used to control the development and evaluation of each feasible solution to the problem Iterative procedure for finding the optimal solution to a linear programming problem



Standard Form of L.P.P.

Characteristics of standard form are All constraints are equations except for non-negativity condition which remain inequalities (0) only RHS element of each constraint is non-negative All variables are non-negative Objective function is of maximization or minimization type

Slack Variable: The non-negative variable that has to be added to a constraint inequality of the form to change it to an equation is called a slack variable. Slack Variable: The non-negative variable that has to be added to a constraint inequality of the form to change it to an equation is called a slack variable.5PREVIOUS HOMECNEXT d e n t i a l onfi


Fundamental Theorem of L.P.P

Theorem: Given a set of m simultaneous linear equations in n unknowns/variables, nm, AX=b, with r(A)=m. If there is a feasible solution X0, then there exists a basic feasible solution. * SOLUTION OF L.P.P PROBLEM: Initial Basic Feasible solution of a L.P.P: Consider a system of m equations in n unknowns x1, x2 - - an, a11 x1 + a12 x2 + - - + a1n xn = b1 a21 x1 + a22 x2 + - - + a2n xn = b2 am1 x1 + am2 x2 + - - + amn xn = bn Where m n



Steps to solve Simplex method:

1. Introduce slack variables (Sis) for type of constraint.

2. Introduce surplus variables (Sis) and artificial variables (Ai) for type of constraint. 3. Introduce only Artificial variable for = type of constraint. 4. Cost (Cj) of slack and surplus variables will be zero and that of artificial variable will be M 5. Find Zj - Cj for each variable. 6. Slack and artificial variables will form basic variable for the first simplex table. Surplus variable will never become basic variable for the first simplex table. 7. Zj = sum of [cost of variable x its coefficients in the constraints Profit or cost coefficient of the variable]. 8. Select the most negative value of Zj - Cj. That column is called key column. The variable corresponding to the column will become basic variable for the next table.

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Steps to solve Simplex method:

9. Divide the quantities by the corresponding values of the key column to get ratios; select the minimum ratio. This becomes the key row. The basic variable corresponding to this row will be replaced by the variable found in step 6. 10. The element that lies both on key column and key row is called Pivotal element. 11. Ratios with negative and value are not considered for determining key row. 12. Once an artificial variable is removed as basic variable, its column will be deleted from next iteration. 13. For maximization problems, decision variables coefficient will be same as in the objective function. For minimization problems, decision variables coefficients will have opposite signs as compared to objective function. 14. Values of artificial variables will always is M for both maximization and minimization problems. 15. The process is continued till all Zj - Cj 0.Confidential

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SIMPLEX ALGORITHM

Simplex algorithm is used to test the optimality of current basic feasible solution of L.P.P. Steps in Simplex Algorithm: Convert to standard form (possibly adding variables) Add slack variables

Pivotchoose most negative reduced cost (identifies new basic variable) choose lowest positive ratio in ratio test eliminate new basic variable from objective and all rows except lowest positive ratio in ration test row Stop if optimal and read solution. Otherwise, pivot again. Etc.9PREVIOUS HOMECNEXT d e n t i a l onfi

Steps to solve Simplex AlgorithmPerform the following steps to solve the simplex algorithm.


step-1

Locate the negative number Form Ratio's User Elementary Row operations Replace variable in pivotal row & column Repeat steps 1 to 4

step-2

step-3

step-4

step-5

step-6

Arrive at an optimal solution10Confidential


Penalty Cost Method or Big-M MethodOverview of Big-M Method: Addition of artificial variables causes violation of the corresponding constraints as they are added to only one side of an equation. The new system is equivalent to the old system of constraints only if the artificial variables are valued at zero. You can make certain modifications in the simplex method to

minimize the error of incorporating the penalty cost in the objective function. This method is called Big M-method or Penalty cost method



Steps to follow in Big-M method:

Step1: Break the last row into two rows

Step 2: Follow the steps from 1 to 3

of Simplex method

Step 3: Artificial variable ceases to be basic variable

Step 4: Delete the last row in the table

Step 5: Arrive at the optimal solution12Confidential


Two Phase Method

Overview of Two Phase Method: The drawback of the penalty cost method is the possible computational error resulting from assigning a very large value to the constant M. To overcome this difficulty, a new method is considered, where the use of M is eliminated by solving the problem in two phases.

Phase I: Phase II



Methodology of Two-phase method:Add slack or subtract surplus variables to put the problem in the form: Minimize CTX subject to AX = b, X >= 0. Add a new variable to the left side of each constraint which originally was >= or = the constant term. Set up a new objective function equal to the sum of the artificial variables.

Put in Standard Form

2. Add Artificial Variables

3. Change Objective Function

4. Perform Phase I

5. Check for Feasibility

6. Insert old Objective Function

7. Perform Phase IIPREVIOUS HOME

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Set up the simplex tableau and pivot to reach a solution where bottom row = 2 x1 >= 0, x2 >= 0, x3 >= 0

subject to: 2 x1 + 4 x2

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PPT-Unit-03-MB0048