OPERATIONS RESEARCHChapter 02 - Simplex method introduction

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INTRODUCTION The standard form of an LPP with n variables and m unknowns is as follows:

Maximize Subject to:

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INTRODUCTION Main features of the LPP standard form:

Maximize or Minimize the objective function. All constraints are equations. All decision variables are nonnegative. The right hand side of each constraint is nonnegative.

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LPP IN MATRIX FORM The above standard LPP can be written in matrix form as: max(min) Z=CX Subject to: Ax=b, x>=0, b>=0. Where,

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EXAMPLE max Subject to:

Can be written in matrix form with:

54321 325 xxxxxZ

5,...,1,07x 43

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5321

4321

ixxxx

xxxx

i

11325,78

,,1014301221

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4

3

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xA

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SLACK AND SURPLUS VARIABLES The above standard form is required in order to start the simplex method. The question is how to convert a given problem from nonstandard form into standard form?, i.e. how to convert an inequality constraint into an equality?.

The answer to this question is to add the so called slack variables and surplus variables.

Back to the example of inspection, which had the constraints: Minimize Subject to:

21 3640 xxZ

0,0453510 8

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2

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x

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SLACK AND SURPLUS VARIABLES Minimize Subject to:

The first constraint is converted to where represents the number of grade 1 inspectors who are not utilized.

The second constraint is converted to Where represents the number of grade 2 inspectors who are not utilized.

are called slack variables. The third constraint is converted to is called a surplus variable. It represents the number of extra pieces inspected over the min. required minimum.

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0,0453510 8

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0,10 442 xxx 4x

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IMPORTANT REMARK If the problem contains a variable which is not restricted in sign, then we do the following: If is unrestricted in sign, then set

The value of (positive or negative) depends on the values of Example: Convert the following problem into a standard LPP:

Solution: 1. Set 2. Multiply the last constraint by -1, to get:3. Add a slack variable ( ) to 1st constraint to become:4. Add a surplus variable ( ) to the 2nd constraint to become:5. Replace by in the original problem , to get the standard form:

1x 321 xxx

1x 32 xandx

edunrestrict x0,x, x-52x-x-3x

2x 7 x: to.32x Z

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xxMax

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IMPORTANT REMARK The standard form:

0,...,,x 5223x- 2 x7 x:.

332x Z

721

5421

75421

65421

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xxxMaximize

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SYSTEMS OF LINEAR EQUATIONS Two systems of linear equations are said to be equivalent if both systems have the same solution set.

The following are called elementary row operations that can be applied to rows of a matrix:1. Exchange any two rows.2. Multiply a row by a nonzero constant.3. Multiply a row by a constant and add the result to another row.

Notice that these operations can also be applied to equations in a system of equations. They are called in this case elementary operations.

Two matrices are said to be row equivalent if one of them can be obtained from the other by a sequence of elementary row operations.

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SYSTEMS OF LINEAR EQUATIONS A matrix is said to be in reduced row-echelon form if it satisfies all four of the following conditions. If there are any rows of all zeros then they are at the bottom of the

matrix. If a row does not consist of all zeros then its first non-zero entry (i.e. the

left most non-zero entry) is a 1.  This 1 is called a leading 1. In any two successive rows, neither of which consists of all zeroes, the

leading 1 of the lower row is to the right of the leading 1 of the higher row.

If a column contains a leading 1 then all the other entries of that column are zero.

A matrix is said to be in row-echelon form if it satisfies items 1- 3 of the reduced row-echelon form definition.

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REMARK Notice that the only real difference between row-echelon form and reduced row-echelon form is that a matrix in row-echelon form is only required to have zeroes below a leading 1 while a matrix in reduced row-echelon from must have zeroes both below and above a leading 1.

A standard form of a system of equations is as follows:

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REMARK The following matrix is called the augmented matrix of the above system:

If two augmented matrices are row equivalent, then they represent two equivalent systems of linear equations.

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REMARK And the following is called is called the coefficient matrix: