the simplex method

27
The Simplex Method

Upload: merrill-cabrera

Post on 03-Jan-2016

46 views

Category:

Documents


2 download

DESCRIPTION

The Simplex Method. Standard Linear Programming Problem. Standard Maximization Problem 1. All variables are nonnegative . 2. All the constraints (the conditions) can be expressed as inequalities of the form: ax + by ≤ c, where c is a positive constant. Illustrating Example (1). - PowerPoint PPT Presentation

TRANSCRIPT

Page 1: The Simplex Method

The Simplex Method

Page 2: The Simplex Method

Standard Linear Programming Problem

Standard Maximization Problem

1. All variables are nonnegative.

2. All the constraints (the conditions) can be expressed as inequalities of the form:

ax + by ≤ c, where c is a positive constant

Page 3: The Simplex Method

Illustrating Example (1)

Maximize the objective function:P(x,y) = 5x + 4ySubject to:x + y ≤ 202x + y ≤ 35-3x + y ≤ 12x ≥ 0y ≥ 0

Page 4: The Simplex Method

Solution

Page 5: The Simplex Method

0

12

35

20

100045

010013

001012

000111

:

:

045

:

123,352,20

::,,

"",Re

45123,352,20

:

:

tsanconstpwvuyx

tablefollowingtheconstructWe

Second

pyx

followsasfunctionobjectivetheofformulatherewriteand

wyxvyxuyx

followsasfollowsaswandvu

riablesvaslacktroducinginbyesinequalitithewrite

yxpandyxyxyx

hadWe

formulaandconditionsgiventherewriteWe

First

Page 6: The Simplex Method

)(2,

,2/35.2/35201/20:,

)

(

.)(

.

.).5(

:)(

:.

:),(

:

0

12

35

20

100045

010013

001012

000111

entryelementpivottheistheiswhichentryingcorrespond

thesoandsmallesttheisquotientTheandarequotientstheHere

rowpivottheislocated

isitwhererowtheandelementpivottheisquotientsmallestthetoingcorrespondentryThe

quotientahavenotdowethennegativeisentrythatIfquotientpositiveaobtainto

columnpivottheinentrypositiveingcorrespondthebyroweachinntconstathedivideWeb

columnxtheiscolumnpivottheThushereiswhch

linethetoleftrowlasttheinentrynegativemostthecontainingcolumntheiswhich

columnpivotthelocateWea

followsasentryelementpivotthelocateWe

Third

tsanconstpwvuyx

Page 7: The Simplex Method

2175

2129

235

25

25

23

23

25

21

21

21

21

)5(,)3(,)(

235

21

21

2

1

1000

0100

0001

0010

0

12

20

100045

010013

0001

000111

0

12

35

20

100045

010013

001012

000111

.;'0

1

423212

2

tsanconstpwvuyx

tsanconstpwvuyx

tsanconstpwvuyx

columnunitatocolumnthatingtransformthusstocolumnpivottheinentriesotherthe

andtoelementpivotthetransformtooperationsrowelementarynecessarytheperformWe

Fourth

RRRRRR

R

Page 8: The Simplex Method

)(2

1,,

2

55

425

5

129

2

5/

2

12935.

2

1/

2

35,5

2

1/

2

5:,

)

(

.)(

.

.).2

3(

:)(

:.

)3(

)4()3(

1000

0100

0001

0010

2

3:,,

.,,

:

2175

2129

235

25

25

23

23

25

21

21

21

21

entryelementpivottheisrowfirstthein

theiswhichentryingcorrespondthesoandsmallesttheisquotientThe

andarequotientstheHere

rowpivottheislocated

isitwhererowtheandelementpivottheisquotientsmallestthetoingcorrespondentryThe

quotientahavenotdowethennegativeisentrythatIfquotientpositiveaobtainto

columnpivottheinentrypositiveingcorrespondthebyroweachinntconstathedivideWeb

columnytheiscolumnpivottheThushereiswhch

linethetoleftrowlasttheinentrynegativemostthecontainingcolumntheiswhich

columnpivotthelocateWea

Step

andstepsepeatR

tsanconstpwvuyx

iswhichrowlasttheinentrynegativeahaveweHere

fourandthreesteprepeatweentriesnegativcontainsstillrowlastthethatafterIf

Fifth

Page 9: The Simplex Method

95

52

15

5

101300

014500

001101

001110

1000

0100

0001

0010

.;

'01

)4(

4131211 )3(,)5(,)(,2

2175

2129

235

25

25

23

23

25

21

21

21

21

tsanconstpwvuyx

tsanconstpwvuyx

columnunitatocolumnthatingtransformthus

stocolumnpivottheinentriesothertheandtoelementpivot

thetransformtooperationsrowelementarynecessarytheperformWe

Step

RRRRRRR

Page 10: The Simplex Method

520,0,5,15max

95max

9552,0,5,15

:

).(

1&

))((

:

.

,

:

wandvuyxwhenoccursvalueThis

ispfunctionobjectivetheofvaluetheTherfore

pandwvuyx

Thus

columnunitawithassiciatedonesthe

ariablesvbasicforthecontainingrowtheinlyingntstaconingcorrespondthe

columnunitaheadingwithassiciatednotonesthe

ariablesvnonbasicallforzero

aresolutinoptialthistoariablesvtheofvaluesingcorrespondThe

reachedbeenhassolutionoptimalthethatmeanswhich

positivearelinetheofleftthetorowlasttheinetriesallNow

Sixth

Page 11: The Simplex Method

What about when all of the constraints (the inequalities) are of

the type “≤ positive constant”But we want to minimize the objective function instead of

maximizing.

Page 12: The Simplex Method

Minimization with “≤” constraintsIllustrating Example (2)

Minimize the objective function:p(x,y) = -2x - 3ySubject to:5x + 4y ≤ 32x + 2y ≤ 10x ≥ 0y ≥ 0

Page 13: The Simplex Method

SolutionLetq(x) = - p(x) = - ( -2x -3y) = 2x + 3yTo minimize p is to maximize q. Thus, we solve the

following standard maximization linear programming problem:

Maximize the objective function:q(x) = 2x + 3ySubject to:5x + 4y ≤ 32x + 2y ≤ 10x ≥ 0y ≥ 0

Page 14: The Simplex Method

Rewriting the inequalities as equations, by introducing the “slack” variables u and v and the formula of the objective function as done in example (1).

5x + 4y ≤ 32 , x + 2y ≤ 10 and q = 2x +3y

Are transformed to:

5x + 4y + u = 32

x + 2y + v = 10

- 2x - 3y + q = 0

Page 15: The Simplex Method

)3,4(),(17,min

:,

)3,4(),(17,max

173,4:

17

.

3

4

100

.....

010

001

:),1(

0

.

10

32

10032

.....

01021

00145

:tableaussimplextheconstructWe

0 q3y2x- 10,v2y x32,u4y 5x

,

67

61

65

61

32

31

yxatiswhichvaluetheattainsp

thenqpSince

yxatiswhichvaluetheattainsq

qandyxThus

antsconstqvuyx

atarriveweexampleinxplainedeasmethodtheApplying

antsconstqvuyx

haveWe

Page 16: The Simplex Method

Standard Linear Programming Problem

Standard Minimization Problem

1. All variables are nonnegative.

2. All the constraints (the conditions) can be expressed as inequalities of the form:

ax + by ≥ c, where c is a positive constant

Page 17: The Simplex Method

Solving

The Standard Minimization Problem

We use the fundamental theorem of Duality

Page 18: The Simplex Method

Illustrating Example (3)

Minimize the objective function:p(x,y) = 6x + 8ySubject to:40x + 10y ≥ 240010x + 15y ≥ 21005x + 15y ≥ 1500x ≥ 0y ≥ 0

Page 19: The Simplex Method

Minimize the objective function: p(x,y) = 6x + 8ySubject to:40x + 10y ≥ 2400, 10x + 15y ≥ 2100 , 5x + 15y ≥ 1500, x ≥ 0 and y ≥ 0We will refer to the above given problem by the primal (original) problem

First: We construct the following table, which we will refer to by the “primal” table:x y constant---------------------------------40 10 240010 15 21005 15 1500---------------------------------6 8

Second: We construct a dual (twin) table from interchanging the rows and columns in the primal table:

x' y' z' constant----------------------------------------------------------- 40 10 5 6 10 15 15 8---------------------------------------------------------2400 2100 1500

Third: We interpret the “dual table” as a standard maximization problem, which will refer to as the “dual problem” or “twin problem” of the “primal problem” or the “original problem”

Miaximoze the objective function: q( x ' , y ' , z ' ) = 2400x' + 2100y' + 1500z'Subject to:40x' + 10y' + 5z' ≤ 6, 10x' + 15y' + 15z' ≤ 8 , x' ≥ 0 and y' ≥ 0, z' ≥ 0

Page 20: The Simplex Method

Fourth: We apply the simplex method explained in example (1) to solve this problem

Maximize the objective function: q(x,y,z) = 2400x' + 2100y' + 1500z'

Subject to:

40x' + 10y' + 5z' ≤ 6, 10x' + 15y' + 15z' ≤ 8 , x' ≥ 0 and y' ≥ 0, z' ≥ 0

4.a.Rewriting the inequalities and the formula of the objective function, with the slack variables being the same x and y (in that order) of the original (minimization) problem :

40x' + 10y' + 5z' + x = 6

10x' + 15y' + 15z' + y = 8

- 2400x' - 2100y' - 1500z‘ + q = 0

4.b. We construct the simplex table for this problem

Page 21: The Simplex Method

1140

.

11203045000

......

010

001

:

..4

0

.

8

6

100150021002400

......

010151510

00151040

0 q z1500 - 2100y' - 2400x' -

8 y 15z' 15y' 10x'

6 x 5z' 10y' 40x'

2513

501

252

501

1011

501

1003

203

antconstqyxzyx

tablefollowingthetoleadwillwhichmethodsimplextheby

dnecessiateoperationsrowelementarytheallperformWec

antconstqyxzyx

Page 22: The Simplex Method

Fifth: We read the solution from the table

4)2(20))2,0(

2,0:max

:

440)2(20)2,0(

:

4maxmin

20:

:""

min)(

4

.

2

3

12001

......

01011

01101

q

yxwhenisq

Checking

p

pofformulathefromanswerthisgetalsocanWe

qofvaluethepofvalueThe

yandxThus

yandxariablesvslacktheunderrowlasttheinappears

problemimizationoriginalprimalthetosolutionThe

antconstqyxyx

Page 23: The Simplex Method

Illustrating Example (4)

Minimize the objective function:p(x,y) = x + 2ySubject to:-2x + y ≥ 1- x + y ≥ 2x ≥ 0y ≥ 0

Page 24: The Simplex Method

Minimize the objective function: p(x,y) = x + 2ySubject to:-2x + y ≥ 1, - x + y ≥ 2 We will refer to the above given problem by the primal (original) problem

First: We construct the following table, which we will refer to by the “primal” table:x y constant----------------------------------2 1 1-1 1 2---------------------------------1 2

Second: We construct a dual (twin) table from interchanging the rows and columns in the primal table:

x' y' constant------------------------------------------- -2 -1 1 1 1 2-----------------------------------------1 2

Third: We interpret the “dual table” as a standard maximization problem, which will refer to as the “dual problem” or “twin problem” of the “primal problem” or the “original problem”

Maximize the objective function: q( x ' , y ‘ ) = x' + 2y' Subject to:-2x' - y' ≤ 1, x' + y' ≤ 2 , x' ≥ 0 and y' ≥ 0

Page 25: The Simplex Method

Fourth: We apply the simplex method explained in example (1) to solve this problem

Maximize the objective function: q( x ' , y ‘ ) = x' + 2y'

Subject to:

- 2x' - y' ≤ 1, x' + y' ≤ 2 , x' ≥ 0 and y' ≥ 0

4.a.Rewriting the inequalities and the formula of the objective function, with the slack variables being the same x and y (in that order) of the original (minimization) problem :

- 2x' - y' ' + x = 1

x' + y' + y = 2

- x' - 2y' + q = 0

4.b. We construct the simplex table for this problem

Page 26: The Simplex Method

4

.

2

3

12001

......

01011

01101

:

..4

0

.

2

1

10021

......

01011

00112

0 q 2y' - x'-

2 y y' x'

1 x ' y' - 2x' -

antconstqyxyx

tablefollowingthetoleadwillwhichmethodsimplextheby

dnecessiateoperationsrowelementarytheallperformWec

antconstqyxyx

Page 27: The Simplex Method

Homework

1. Using the simplex method, maximize: p = x + (6/5)y subject to:2x + y ≤ 180 , x + 3y ≤ 300 , x ≥ 0 , y ≥ 0Solution: p(48,84) = 148.8

2. Minimize: p(x,y) = - 5x - 4y Subject to: x + y ≤ 20 , 2x + y ≤ 35 , -3x + y ≤ 12 , x ≥ 0y ≥ 0Solution: p(15,5) = - 95

3. Using the dual theorem, minimize: p = 3x + 2y subject to:8x + y ≥ 80 , 8x + 5y ≥ 240 , x + 5y ≥ 100, x ≥ 0 , y ≥ 0Solution: p(20,16) = 92Maximize the objective function: