optimization simplex method introduction

Algebraic Solution of LPPs - Simplex Method

To solve an LPP algebraically, we first put it in the standard form. This means all decision variables are nonnegative and all constraints (other than the nonnegativity restrictions) are equations with nonnegative RHS.

Converting inequalities into equations

21 32 xxz

Subject to

0,

623

63

21

21

21

xx

xx

xx

Maximize

Consider the LPP

We make the ≤ inequalities into equations by adding to each inequality a “slack” variable (which is nonnegative). Thus the given LPP can be written in the equivalent form

21 32 xxz

Subject to

0,,,

623

63

2121

221

121

ssxx

sxx

sxx

are slack variables.21, ss

Maximize

Thus we seem to have complicated the problem by introducing two more variables; but then we shall see that this is easier to solve. This is one of the “beauties” in mathematical problem solving.

The ≥ inequalities are made into equations by subtracting from each such inequality a “surplus” (non-negative) variable.

Thus the LPP

21 32 xxz Subject to

0,

223

63

21

21

21

xx

xx

xx

Maximize

is equivalent to the LPP

21 32 xxz

Subject to

0,,,

223

63

2121

221

121

ssxx

sxx

sxx

is a slack variable; is a surplus variable.2s1s

Maximize

If in a constraint, the RHS constant is negative, we make it positive by multiplying the constraint by -1.

Thus the LPP

21 32 xxz

Subject to

0,

223

63

21

21

21

xx

xx

xx

Maximize


21 32 xxz

Subject to 1 2

1 2

1 2

3 6

3 2 2

, 0

x x

x x

x x

Maximize

Its standard form is the LPP

21 32 xxz

Subject to

0,,,

223

63

2121

221

121

ssxx

sxx

sxx

Maximize


Dealing with unrestricted variables

If, in an LPP, a decision variable xi is unrestricted (in sign) i.e. it can take positive as well as negative values, then we can, by writing i i ix x x ,i ix x

are (defined below and are) nonnegative, make the LPP into an equivalent LPP where all the decision variables are ≥ 0.

Note:| |

;2i i

i

x xx

if 0 and otherwisei i i ix x x x

where

| |

2i i

i

x xx

Thus the LPP

Maximize 21 3xxz Subject to

1 2

1 2

1 2

2

4

unrestricted, 0

x x

x x

x x


211 3xxxz

Subject to

1 1 2 1

1 1 2 2

1 1 2 1 2

2

4

, , , , 0

x x x s

x x x s

x x x s s

Maximize

Basic variables, Basic feasible Solutions

Consider an LPP (in standard form) with m constraints and n decision variables. We assume m ≤ n. We choose n –m variables and set them equal to zero. Thus we will be left with a system of m equations in m variables. If this mm square system has a unique solution, this solution is called a basic solution. If further if it is feasible, it is called a Basic Feasible Solution (BFS).

The n-m variables set to zero are called nonbasic and the m variables which we are solving for are known as basic variables. Thus a basic solution is of the form x = (x1, x2, …, xn) where n-m “components” are zero and the remaining m components form the unique solution of the square system (formed by the m constraint equations). Note that we may have a maximum of basic solutions.

n

m

Consider the LPP:

Maximize 21 32 xxz

Subject to

0,

623

63

21

21

21

xx

xx

xx

CThis is equivalent to the LPP (in standard form)

Maximize 21 32 xxz

Subject to

0,,,

623

63

2121

221

121

ssxx

sxx

sxx


Nonbasic Nonbasic (zero) (zero) variablesvariables

Basic Basic variablesvariables

Basic Basic solutionsolution

AssocAssoc-iated -iated corner corner pointpoint

Feasible?Feasible?Object-Object-

ive value, ive value, z z

(0,0,6,6)(0,0,6,6) OO YesYes 00

(0,2,0,2)(0,2,0,2) BB YesYes 66

(0,3,-3,0)(0,3,-3,0) EE NoNo --

(6,0,0, -12)(6,0,0, -12) DD NoNo --

(2,0,4,0)(2,0,4,0) AA YesYes 44

CC YesYes48/748/7

OptimalOptimal)0,0,

7

12,

7

6(

),,,( 2121 ssxx

),( 21 xx

),( 11 sx

),( 21 sx

),( 12 sx

),( 22 sx

),( 21 ss ),( 21 xx

),( 11 sx

),( 21 sx

),( 12 sx

),( 22 sx

),( 21 ss

Graphical solution of the above LPP

x1

x2

O A D

B

E

C

(2,0) (6,0)

(0,2)

(0,3) (6/7, 12/7)Optimal point

(0,0)

Thus every Basic Feasible Solution corresponds to a corner(=vertex) of the set SF of all feasible solutions.

SF

Consider the LPP:

Maximize 21 3xxz

Subject to 1 2

1 2

1 2unrestricted

2

4

, 0

x x

x x

x x

Question 6 (Problem set 3.2A – Page 79)

This is equivalent to the LPP(in standard form)

Maximize 211 3xxxz

Subject to

0,,,,

4

2

21211

2211

1211

ssxxx

sxxx

sxxx




Assoc-Assoc-iated iated corner corner pointpoint

Feasible?Feasible? Objective Objective value, zvalue, z

(0,0,0,2,4)(0,0,0,2,4) OO YesYes 00

(0,0,2,0,2)(0,0,2,0,2) BB YesYes 66

(0,0,4,-2,0)(0,0,4,-2,0) EE NoNo --

(0,-2,0,0,6)(0,-2,0,0,6) -- NoNo --

(0,4,0,6,0)(0,4,0,6,0) DD YesYes -4-4

(0,1,3,0,0)(0,1,3,0,0) CC YesYes 88

),,,,( 21211 ssxxx

),,( 111 sxx

),,( 121 sxx

),,( 211 xxx

),,( 211 sxx

),,( 221 sxx

),,( 211 ssx

),( 21 ss

),( 21 xx

),( 11 sx

),( 21 sx

),( 12 sx

),( 22 sx

Optimal value




Associ-Associ-ated ated corner corner pointpoint

FeasibleFeasible??

Objective Objective value, zvalue, z

(2,0,0,0,6)(2,0,0,0,6) AA YesYes 22

(-4,0,0,6,0)(-4,0,0,6,0) -- NoNo --

(-1,0,3,0,0)(-1,0,3,0,0) -- NoNo --

N0 N0 SolutionSolution -- -- --

),,,,( 21211 ssxxx

),,( 221 sxx

),,( 212 ssx

),,( 211 ssx

),,( 121 sxx

),( 11 xx

),( 21 xx

),( 11 sx

),( 21 sx

Hence note that the number of Basic Solutions can be less than n

m

(-1,3)z maximum 8 at

0

B

E

D A

C

Direction of increasing z

21 3xxz

Example: Convert the following optimization problem into a LPP:

Maximize

1 2 1 2max{| 2 3 |, | 3 7 |}z x x x x

Subject to1 2

1 2

1 2

2

4

, 0

x x

x x

x x

Note that here the objective function is NOT linear. Let us put

1 2 1 2max{| 2 3 |, | 3 7 |}y x x x x

Hence 1 2 1 2| 2 3 | and | 3 7 |y x x y x x

Which is equivalent to

1 2 1 22 3 , (2 3 )y x x y x x

1 2 1 23 7 , 3 7y x x y x x

Hence the given optimization problem is equivalent to the LPP:

Maximize z ySubject to

1 2

1 2

2

4

x x

x x

1 2 1 22 3 0, 2 3 0x x y x x y

1 2 1 2

1 2

3 7 0, 3 7 0.

, , 0

x x y x x y

x x y

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