Linear Programming _Simplex (1)

7/28/2019 Linear Programming _Simplex (1)

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To accompany Quantitative Analysis

for Management, 8e

by Render/Stair/Hanna9-1

2003 by Prentice Hall, Inc.

Upper Saddle River, NJ 07458

LinearLinearProgramming:Programming:

The SimplexThe SimplexMethodMethod


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Flair FurnitureFlair Furniture

CompanyCompany

Maximize:Objective: 2157 XX +

Hours Required to Produce One Unit

DepartmentX1

Tables

X2Chairs

Available

Hours This

Week

Carpentry

Painting/Varnishing

4

2

3

1

240

100

Profit/unit

Constraints:

$7 $5

)varnishing&(painting

10012 21 + XX

)(carpentry2403421+ XX


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Feasible Region & CornerFeasible Region & CornerPointsPoints

Num

berofCha

irs

100

80

60

40

20

0 20 40 60 80 100X

X2

Number of Tables

B = (0,80)

C = (30,40)

D = (50,0)

Feasible

Region

2403421+ XX

1001211+XX


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Flair Furniture -Flair Furniture -Adding SlackAdding Slack

VariablesVariables

)varnishing&(painting10012 21 + XX

)(carpentry24034 21 + XX

Constraints:

Constraints with Slack Variables

)varnishing&(painting

)(carpentry

10012

24034

221

121

=++

=++

SXX

SXX

2157 XX +

Objective Function

Objective Function with Slack

Variables2121

0057 SSXX +++


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Flair Furnitures InitialFlair Furnitures InitialSimplex TableauSimplex Tableau

Profit

per

Unit

olumnProd.

Mix

Column

RealVariables

Columns Slack

Variables

Columns

Constant

Column

Cj

Solution

Mix X1

X2

S1

S2

Quantity

$7 $5 $0 $0Profit

per

unit row

4 3 1 0

2 1 0 1

$0 $0 $0 $0

$7 $5 $0 $0

$0

$0

S1

S2

Zj

Cj -

Zj

240

100

$0

$0

Constrainequation

rows

GrossProfit

rowNet

Profitrow


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Pivotal row, column, pivotPivotal row, column, pivotno., identified in the Initialno., identified in the Initial

Simplex TableauSimplex Tableau

Cj

Solution

Mix X1 X2 S1 S2 Quantity

$7 $5 $0 $0

4 3 1 0

2 1 0 1

$0 $0 $0 $0

$7 $5 $0 $0

$0

$0

S1

S2

Zj

Cj -Zj

240

100

$0

Pivotal row

(smallest ratio)Pivotal number

Pivotal column (largest Cj-Zj)

ratio

60

50


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Second Simplex TableauSecond Simplex Tableau

Cj

Solution


$7 $5 $0 $0

0 1 1 -2

1 1/2 0 1/2

$7 $7/2 $0 $7/2

$0 $3/2 $0 -$7/2

$0

$7

S1

X1

Zj

Cj -Zj

40

50

$350

Pivotal row

(smallest ratio)

Pivotal number

Pivotal column (largest Cj-Zj)

ratio

40

100


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Third Simplex TableauThird Simplex Tableau

Cj

Solution


$7 $5 $0 $0

0 1 1 -2

1 0 -1/2 3/2

$7 $5 $3/2 $1/2

$0 $0 -$3/2 -$1/2

$5

$7

X2

X1

Zj

Cj -Zj

40

30

$410

Final Tableau since Cj-Zj has no positive values


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a cu at ng t e ewa cu a ng e ew 11Row for Flairs ThirdRow for Flairs Third

TableauTableau

= - x1

0

3/2

-1/2

30

1

1/2

1/2

0

50

(1/2)

(1/2)

(1/2)

(1/2)

(1/2)

(0)

(1)

(-2)

(1)

(40)

= - x= - x

= - x

= - x

=

rowX

newin

number

ingCorrespond

number

pivot

above

Number

rowX

oldin

Number

RowX

Newin

Number

ii


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Simplex Steps forSimplex Steps for

MaximizationMaximization

1. Choose the variable with the greatest

positive Cj - Zj to enter the

solution.

2. Determine the row to be replaced byselecting that one with the smallest

(non-negative) quantity-to-pivot-

column ratio.

3. Calculate the new values for thepivot row.

4. Calculate the new values for the

other row(s).

5. Calculate the Cj and Cj - Zj values for

this tableau. If there are any Cj - Zj

values greater than zero, return to

Step 1.


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Surplus & ArtificialSurplus & Artificial

VariablesVariablesConstraints

Constraints-Surplus & ArtificialVariables

9003025

2108105

21

321

=+

++

XX

XXX

9003025

2108105

221

11321

=++

=+++

AXX

ASXXX

Objective Function

321795 XXX ++:Min

2113210795 MAMASXXX +++++:Min

Objective Function-Surplus & Artificial

Variables


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Simplex Steps forSimplex Steps for

MinimizationMinimization

1. Choose the variable with the greatest

negative Cj - Zj to enter the solution.

2. Determine the row to be replaced byselecting that one with the smallest

(non-negative) quantity-to-pivot-

column ratio.

3. Calculate the new values for the pivot

row.

4. Calculate the new values for the other

row(s).5. Calculate the Cj and Cj - Zj values for

this tableau. If there are any Cj - Zj

values less than zero, return to Step 1.


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Special CasesSpecial Cases

UnboundednessUnboundedness

Pivot

Column

Cj 6 9 0 0

Sol

Mix

X1

X2

S1

S2

Qty

X1 -1 1 2 0 30

S1 -2 0 -1 1 10

Zj -9 9 18 0 270

Cj - Zj 15 0 -18 0

all ratios -ve


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Degeneracy (redundancy)Degeneracy (redundancy)

Pivot Column

Cj 5 8 2 0 0 0Solution

MixX1 X2 X3 S1 S2 S3 Qty

8 X2 1/4 1 1 -2 0 0 10

0 S2 4 0 1/3 -1 1 0 20

0 S3 2 0 2 2/5 0 1 10

Zj 2 8 8 16 0 0 80

Cj-Zj 3 0 -6 -16 0 0

Tied ratios


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Multiple OptimalMultiple Optimal

Cj 3 2 0 0

Sol

Mix

X1

X2

S1

S2

Qty

2 X1 3/2 1 1 0 6

0 S2 1 0 1/2 1 3

Zj 3 2 2 0 12

Cj - Zj 0 0 -2 0

Cj-Zj=0, real v not in sol mix, optimal


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Sensitivity AnalysisSensitivity Analysis

High Note Sound CompanyHigh Note Sound Company

6013

8042

12050

21

21

21

+

+

+

XX

XX

XX

:toSubject

:Max

Elect time (hrs)

Audio-tech time (hrs)


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Sensitivity AnalysisSensitivity Analysis



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Simplex SolutionSimplex Solution


Cj 50 120 0 0Sol

Mix

X1 X2 S1 S2 Qty

120 X2

1/2 1 1/4 0 20

0 S2 5/2 0 -1/4 1 40

Zj 60 120 30 0 2400

Cj

-

Zj

-10 0 -30 0


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Nonbasic ObjectiveNonbasic Objective

Function CoefficientsFunction Coefficients

Cj 50 120 0 0

Sol

Mix

X1 X2 S1 S2 Qty

120 X2 1/2 1 1/4 0 20

0 S2 5/2 0 -1/4 1 40

Zj 60 120 30 0 2400

Cj Zj -10 0 -30 0


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Basic Objective FunctionBasic Objective Function

CoefficientsCoefficients

Cj 50 120 0 0

Sol

Mix

X1 X2 S1 S2 Qty

120

+

X1 1/2 1 1/4 0 20

0 S2 5/2 0 -1/4 1 40

Zj 60+

/2

120

+

30+

/4

0 2400

+20

Cj - Zj -10-/2 0 -30-/4 0


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Simplex SolutionSimplex Solution


Objective increases by 30 if 1

additional hour of electricians time

is available.

Cj 50 120 0 0

Sol

Mix

X1 X2 S1 S2 Qty

X1 1 1/4 0 20

S2 5/2 0 -1/4

1 40

Zj 60 120 30 0 40

Cj -

Zj

0 0 -30 0 2400

Optimal tableau, Cj-Zj


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Shadow PricesShadow Prices

Shadow Price: Value of One

Additional Unit of a Scarce

Resource

Found in Final Simplex Tableau

in C-Z Row

Negatives of Numbers in Slack

Variable Column


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Steps to Form the DualSteps to Form the Dual

To form the Dual:

If the primal is max., the dual is min.,

and vice versa.

The right-hand-side values of the primalconstraints become the objective

coefficients of the dual.

The primal objective function

coefficients become the right-hand-side

of the dual constraints.

The transpose of the primal constraint

coefficients become the dual constraintcoefficients.

Constraint inequality signs are reversed.


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Primal & DualPrimal & Dual

Primal: Dual

6013

8042

21

21

+

+

XX

XX

Subject to:

12014

5032

21

21

+

+

UU

UU

Subject to:

1205021

+XX:Max

608021

+

UU:Min


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Comparison of the PrimalComparison of the Primal

and Dual Optimal Tableausand Dual Optimal Tableaus

PrimalsOptimalSolution

DualsOptimalSolution

CjSolution

Mix

Quantity

$7

$5

X2

S2

Zj

Cj -Zj

20

40

$2,400

X1 X2 S1 S2

$50 $120 $0 $0

1/2 1 1/4 0

5/2 0 -1/4 1

60 120 30 0

-10 0 -30 0

CjSolution

Mix

Quantity

$7

$5

U1

S1

Zj

Cj -Zj

30

10

$2,400

X1 X2 S1 S2

80 60 $0 $0

1 1/4 0 -1/4

0 -5/2 1 -1/2

80 20 0 -20

$0 40 0 20

A1 A2

M M

0 1/2

-1 1/2

0 40

M M-40


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Linear Programming _Simplex (1)