1. Big M Method Linear Programming Pdf Free
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3. Big M Method Linear Programming Pdf

Big m method Linear Programming Example

Big M Method Linear Programming Pdf Free

So far, we have seen the linear programming constraints with less than type. We come across problems with ‘greater than’ and ‘equal to’ type also. Each of these types must be converted as equations. In case of ‘greater than’ type, the constraints are rewritten with a negative surplus variable s₁ and by adding an artificial variable a.

Download as PDF, TXT or read online from Scribd. Flag for inappropriate content. Artificial Starting Solution The Big M Method. The linear programming problem in which all constraints are (≤) with. Artificial Starting Solution The Big M Method The Big M Method Calculations First and second simplex tableau are as. Designated by –M for maximization problems (+M for minimizing problem), where M > 0. Step 3 – In the last, use the artificial variables for the starting solution and proceed with the usual simplex routine until the optimal solution is obtained. 6.2 Worked Examples Lecture 6 Linear programming: Artificial variable technique: Big - M method 1. Big M for a max (min) Linear Programming problem: Step 1. Introduce artificial variables in each row (with no basic variable). 2 phase method for a Linear Programming problem: Step 1. Introduce artificial variables in each row (with no basic variable). Objective for phase 1: minw = a1 +a2 +.+a m. “clean-up” the. Extra Problems for Chapter 3. Linear Programming Methods 20. (Big-M Method ) An alternative to the two-phase method of finding an initial basic feasible solution by minimizing the sum of the artificial variables, is to solve a single linear program in which the objective function is. Linear Programming: Penn State Math 484 Lecture Notes Version 1.8.3 Christopher Gri n. Simplex Method{Tableau Form78 5. Identifying Unboundedness81 6. Identifying Alternative Optimal Solutions84. Arti cial Variables91 2. The Two-Phase Simplex Algorithm95 3. The Big-M Method98 4. The Single Arti cial Variable Technique102 5. Problems that. A-2 Module A The Simplex Solution Method T he simplex method,is a general mathematical solution technique for solving linear programming problems. In the simplex method, the model is put into the form of a table, and then a number of mathematical steps are performed on the table. In operations research, the Big M method is a method of solving linear programming problems using the simplex algorithm. The Big M method extends the power of the simplex algorithm to problems that contain 'greater-than' constraints.

Artificial variables are simply used for finding the initial basic solutions and are thereafter eliminated. In case of an ‘equal to’ constraint, just add the artificial variable to the constraint. The co-efficient of artificial variables a₁, a₂,….. are represented by a very high value M, and hence the method is known as BIG-M Method.

Big m method Minimization Problem

The Big m method minimization problem are explained below

Example : Solve the following LPP using Big M Method.

Minimize Z = 3x₁ + x₂
Subject to constraints
4x₁ + x₂ = 4 ....................(i)
5x₁ + 3x₂≥ 7 ....................(ii)
3x₁ + 2x₂≤ 6 ....................(iii)
where x₁ , x₂≥0

Solution: Introduce slack and auxiliary variables to represent in the standard form. Constraint 4x₁ + x₂ = 4 is introduced by adding an artificial variable a₁, i.e., 4x₁ + x₂ + a₁ = 4 Constraint, 5x₁ + 3x₂≥ 7 is converted by subtracting a slack s₁ and adding an auxiliary variable a₂.

5x₁+ 3x₂– s₁ + a₂ = 7

Constraint 3x₂ + 2x₂≤ 6 is included with a slack variable s₂

3x₂ + 2x₂ + s₂ = 6

The objective must also be altered if auxiliary variables exist. If the objective function is minimization, the co-efficient of auxiliary variable is +M (and -M, in case of maximization)

The objective function is minimization,

Minimize Z = 3x₁+ x₂ + 0s₁+ 0s₂+ Ma₁+ Ma₂

z_min = 3x₁ + x₂+ Ma₁+ Ma₂

Linear Programming Ppt

The initial feasible solution is (Put x₁, x₂, s₁ = 0)

a₁ = 4
a₂ = 7
s₂ = 6

Establish a table as shown below and solve:

Simplex Table

The solution is,

x₁ = 5/7 or 0.71
x₂ = 8/7 or 1.14
z_min = 3 x 5 / 7 + 8/7
= 23/7 or 3.29

In operations research, the Big M method is a method of solving linear programming problems using the simplex algorithm. The Big M method extends the simplex algorithm to problems that contain 'greater-than' constraints. It does so by associating the constraints with large negative constants which would not be part of any optimal solution, if it exists.

Algorithm[edit]

The simplex algorithm is the original and still one of the most widely used methods for solving linear maximization problems. However, to apply it, the origin (all variables equal to 0) must be a feasible point. This condition is satisfied only when all the constraints (except non-negativity) are less-than constraints and with a positive constant on the right-hand side. The Big M method introduces surplus and artificial variables to convert all inequalities into that form. The 'Big M' refers to a large number associated with the artificial variables, represented by the letter M.

The steps in the algorithm are as follows:

1. Multiply the inequality constraints to ensure that the right hand side is positive.
2. If the problem is of minimization, transform to maximization by multiplying the objective by -1
3. For any greater-than constraints, introduce surplus and artificial variables (as shown below)
4. Choose a large positive Value M and introduce a term in the objective of the form -M multiplying the artificial variables
5. For less-than or equal constraints, introduce slack variables so that all constraints are equalities
6. Solve the problem using the usual simplex method.

For example, x + y ≤ 100 becomes x + y + s₁ = 100, whilst x + y ≥ 100 becomes x + y − s₁ + a₁ = 100. The artificial variables must be shown to be 0. The function to be maximised is rewritten to include the sum of all the artificial variables. Then row reductions are applied to gain a final solution.

The value of M must be chosen sufficiently large so that the artificial variable would not be part of any feasible solution.

For a sufficiently large M, the optimal solution contains any artificial variables in the basis (i.e. positive values) if and only if the problem is not feasible.

Other usage[edit]

When used in the objective function, the Big M method sometimes refers to formulations of linear optimization problems in which violations of a constraint or set of constraints are associated with a large positive penalty constant, M.

When used in the constraints themselves, one of the many uses of Big M, for example, refers to ensuring equality of variables only when a certain binary variable takes on one value, but to leave the variables 'open' if the binary variable takes on its opposite value. One instance of this is as follows: for a sufficiently large M and z binary variable (0 or 1), the constraints

${\displaystyle x-y\le Mz}$

${\displaystyle x-y\ge -Mz}$

Big M Method Linear Programming Pdf

ensure that when ${\displaystyle z=0}$ then ${\displaystyle x=y}$. Otherwise, when ${\displaystyle z=1}$, then ${\displaystyle -M\le x-y\le M}$, indicating that the variables x and y can have any values so long as the absolute value of their difference is bounded by ${\displaystyle M}$ (hence the need for M to be 'large enough.')

See also[edit]

• Two phase method (linear programming) another approach for solving problems with >= constraints
• Karush–Kuhn–Tucker conditions, which apply to Non-Linear Optimization problems with inequality constraints.

References and external links[edit]

Bibliography

• Griva, Igor; Nash, Stephan G.; Sofer, Ariela. Linear and Nonlinear Optimization (2nd ed.). Society for Industrial Mathematics. ISBN978-0-89871-661-0.

Discussion

• Simplex – Big M Method, Lynn Killen, Dublin City University.
• The Big M Method, businessmanagementcourses.org
• The Big M Method, Mark Hutchinson