simplex linear programming i. concept ii. model template iii. class example iv. procedure v....

Simplex Linear ProgrammingSimplex Linear Programming

I. ConceptII. Model TemplateIII. Class ExampleIV. ProcedureV. Interpretation

MAXIMIZATIONMETHOD

Applied Management Science for Decision Making, 2e Applied Management Science for Decision Making, 2e © 2014 Pearson Learning Solutions Philip A. Vaccaro , PhD© 2014 Pearson Learning Solutions Philip A. Vaccaro , PhD

Resource Planning

and Allocation Management

MGMT E-5050


* * DECISION VARIABLES MIGHT BE PRODUCTS BEING CONSIDERED FOR PRODUCTION

Most real-life LP problems have more than twodecision variables*, and thus are too large for the

simple graphical solution procedure.

In simplex LP problems the optimal solution will lie at a corner point of a multi-sided, multi-

dimensional figure called an n - dimensional polyhedron that represents the feasible region.

THE FEASIBLE REGION WOULD BE SIMILARTO THE SURFACE OF A DIAMOND


The simplex method examines the corner points in a systematic fashion, using basic algebraic concepts. The same set of procedures is repeated time after time until an optimal solution is reached.

Each repetition, or iteration, increases the value of the objective function so that we are always moving closer to the optimal solution.


The simplex method yields valuable economic information in addition to the optimal solution.

The manual computations must be mastered in order to successfully employ the software pro- grams, and to interpret the computer printouts.

WHY LEARN THE MANUAL COMPUTATIONS?

Problem StatementProblem Statement

Black and White

Color

A firm produces two different types of television sets:

Three resources are required to produce those televisions:

Chassis 24 units

Labor 160 hours

Color Tubes 10 units

Problem StatementProblem Statement

TELEVISION RESOURCE REQUIREMENTSAND

PROFIT MARGINS

Television Chassis Labor Hours Color Tubes Unit ProfitBlack + White 1 5 0 $6.00

Color 1 10 1 $15.00


Let X1 = BLACK AND WHITE TELEVISIONSLet X2 = COLOR TELEVISIONS

Objective Function:

Maximize Z = 6X1 + 15X2

subject to: 1X1 + 1X2 =< 24 chassis 5X1 + 10X2 =< 160 labor hours 0X1 + 1X2 =< 10 color tubes

X1 , X2 => 0

THE MODELTHE MODEL

Simplex Linear ProgrammingSimplex Linear ProgrammingCONVERSION TO LINEAR EQUALITIESCONVERSION TO LINEAR EQUALITIES

1X1 + 1X2 + 1SS11 = 24 chassis

5X1 + 10X2 + 1SS22 = 160 labor hours

0X1 + 1X2 + 1SS33 = 10 color tubes

SLACK VARIABLES: : S1S1 = CHASSIS , , S2S2 = = LABOR HOURS , , S3S3 = COLOR TUBES = COLOR TUBES

ADDING A SLACK VARIABLE TO EACH CONSTRAINTAND SETTING IT EQUAL TO THE RIGHT – HAND SIDE

Simplex Linear ProgrammingSimplex Linear ProgrammingCONVERSION TO LINEAR EQUALITIES SUITABLE FORCONVERSION TO LINEAR EQUALITIES SUITABLE FOR

INCLUSION IN THE SIMPLEX MATRIXINCLUSION IN THE SIMPLEX MATRIX

1X1 + 1X2 + 1S1 + 00S2 + 00S3 = 24 chassis

5X1 + 10X2 + 00S1 + 1S2 + 00S3 = 160 labor hours

00X1 + 1X2 + 00S1 + 00S2 + 1S3 = 10 color tubes

REWRITE EACH CONSTRAINT TO REFLECT THE PRESENCEPRESENCE ORABSENCEABSENCE OF ALL VARIABLES IN THE PROBLEM

Simplex Linear ProgrammingSimplex Linear Programming11stst FEASIBLE SOLUTION FEASIBLE SOLUTION

TRADITIONALLY THE FIRST FEASIBLE SOLUTIONPRODUCES NO PRODUCT AND HAS ALL

RESOURCES INTACT

XX11BLACK AND

WHITE TVs 0

XX22 COLOR TVs 0

SS11 CHASSIS 24

SS22 LABOR HOURS 160

SS33 COLOR TUBES 10

The 1The 1stst Feasible Solution Feasible Solution

CjCj

BasisBasis QtyQty XX11 XX22 SS11 S2 SS33

SS11

SS22

SS33

ZjZj

Cj-ZjCj-Zj ----------

THE THE BASISBASIS OR OR MIX MIX COLUMNCOLUMN

SHOWS ALL VARIABLES GREATER THAN ZERO IN THE CURRENT SOLUTIONSHOWS ALL VARIABLES GREATER THAN ZERO IN THE CURRENT SOLUTION

Toward 1Toward 1stst Feasible Solution Feasible Solution

CjCj


SS11 2424 11 11 11 00 00

SS22 160160 55 1010 00 11 00

SS33 1010 00 11 00 00 11

ZjZj

Cj-ZjCj-Zj

INSERTING LINEAR EQUALITIES INTO THE MATRIXINSERTING LINEAR EQUALITIES INTO THE MATRIX

RIGHT-HANDRIGHT-HANDSIDES ARESIDES AREENTEREDENTEREDINTO THEINTO THEQUANTITYQUANTITYCOLUMNCOLUMN


CjCj


S1S1 2424 11 11 11 00 00

ZjZj

Cj-ZjCj-Zj


1X1X11 + 1X + 1X22 + 1S + 1S11 + 0S + 0S22 + 0S + 0S33 = 24 = 24


CjCj


S2S2 160160 55 1010 00 11 00

ZjZj

Cj-ZjCj-Zj


5X5X11 + 10X + 10X22 + 0S + 0S11 + 1S + 1S22 + 0S + 0S33 = 160 = 160


CjCj


S3S3 1010 00 11 00 00 11

ZjZj

Cj-ZjCj-Zj


0X0X11 + 1X + 1X22 + 0S + 0S1 1 + 0S+ 0S22 + 1S + 1S33 = 10 = 10


CjCj


SS11 2424 11 11 11 00 00

SS22 160160 55 1010 00 11 00

SS33 1010 00 11 00 00 11

ZjZj

Cj-ZjCj-Zj



CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 2424 11 11 11 00 00

$0.$0. SS22 160160 55 1010 00 11 00

$0.$0. SS33 1010 00 11 00 00 11

ZjZj

Cj-ZjCj-Zj ----------

THE THE ““Cj”Cj” oror CONTRIBUTION MARGINCONTRIBUTION MARGIN

THE GROSS PROFIT PER UNIT FOR ALL VARIABLES IN THE PROBLEM.THE GROSS PROFIT PER UNIT FOR ALL VARIABLES IN THE PROBLEM.SURPLUSSURPLUS AND AND SLACKSLACK VARIABLES HAVE $0.00 Cj’s BY DEFINITION. VARIABLES HAVE $0.00 Cj’s BY DEFINITION.


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 2424 11 11 11 00 00

$0.$0. SS22 160160 55 1010 00 11 00

$0.$0. SS33 1010 00 11 00 00 11

ZjZj

Cj-ZjCj-Zj


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 2424 11 11 11 00 00

$0.$0. SS22 160160 55 1010 00 11 00

$0.$0. SS33 1010 00 11 00 00 11

ZjZj $0.$0.

Cj-ZjCj-Zj

COMPUTING THE Zj ROWCOMPUTING THE Zj ROW

$0.$0.

$0.$0.

$0.$0.

MULTIPLY EACH SLACK VARIABLE’S Cj BY ITS QUANTITY AND ADD THE RESULTSMULTIPLY EACH SLACK VARIABLE’S Cj BY ITS QUANTITY AND ADD THE RESULTS


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 2424 11 11 11 00 00

$0.$0. SS22 160160 55 1010 00 11 00

$0.$0. SS33 1010 00 11 00 00 11

ZjZj $0.$0. $0.$0.

Cj-ZjCj-Zj


$0.$0.

$0.$0.

$0.$0.

THEN MULTIPLY EACH SLACK VARIABLE’S Cj BY ITS VARIABLE COEFFICIENTS AND ADD THE RESULTSTHEN MULTIPLY EACH SLACK VARIABLE’S Cj BY ITS VARIABLE COEFFICIENTS AND ADD THE RESULTS


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 2424 11 11 11 00 00

$0.$0. SS22 160160 55 1010 00 11 00

$0.$0. SS33 1010 00 11 00 00 11

ZjZj $0.$0. $0.$0. $0.$0.

Cj-ZjCj-Zj


$0.$0.

$0.$0.

$0.$0.



CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.

BasisBasis QtyQty XX11 X2X2 SS11 S2 SS33

$0.$0. SS11 2424 11 11 11 00 00

$0.$0. SS22 160160 55 1010 00 11 00

$0.$0. SS33 1010 00 11 00 00 11

ZjZj $0.$0. $0.$0. $0.$0. $0.$0.

Cj-ZjCj-Zj


$0.$0.

$0.$0.

$0.$0.



CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.

BasisBasis QtyQty XX11 XX22 SS11 SS22 SS33

$0.$0. SS11 2424 11 11 11 00 00

$0.$0. SS22 160160 55 1010 00 11 00

$0.$0. SS33 1010 00 11 00 00 11

ZjZj $0.$0. $0.$0. $0.$0. $0.$0. $0.$0.

Cj-ZjCj-Zj


$0.$0.

$0.$0.

$0.$0.



CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 2424 11 11 11 0 00

$0.$0. SS22 160160 55 1010 00 1 00

$0.$0. SS33 1010 00 11 00 0 11

ZjZj $0.$0. $0.$0. $0.$0. $0.$0. $0. $0.$0.

Cj-ZjCj-Zj


$0.$0.

$0.$0.

$0.$0.



CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 2424 11 11 11 00 00

$0.$0. SS22 160160 55 1010 00 11 00

$0.$0. SS33 1010 00 11 00 00 11

ZjZj $0.$0. $0.$0. $0.$0. $0.$0. $0.$0. $0.$0.

Cj-ZjCj-Zj

THE COMPUTED Zj ROWTHE COMPUTED Zj ROW


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 2424 11 11 11 00 00

$0.$0. SS22 160160 55 1010 00 11 00

$0.$0. SS33 1010 00 11 00 00 11

ZjZj $0.$0. $0.$0. $0.$0. $0.$0. $0.$0. $0.$0.

Cj-ZjCj-Zj


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 2424 11 11 11 00 00

$0.$0. SS22 160160 55 1010 00 11 00

$0.$0. SS33 1010 00 11 00 00 11

ZjZj $0.$0. $0.$0. $0.$0. $0.$0. $0.$0. $0.$0.

Cj-ZjCj-Zj ---------- $6.$6.

COMPUTING THE Cj – Zj ROWCOMPUTING THE Cj – Zj ROW

ON A COLUMN BY COLUMN BASIS, SUBTRACT Zj FROM Cj ON A COLUMN BY COLUMN BASIS, SUBTRACT Zj FROM Cj


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 2424 11 11 11 00 00

$0.$0. SS22 160160 55 1010 00 11 00

$0.$0. SS33 1010 00 11 00 00 11

ZjZj $0.$0. $0.$0. $0.$0. $0.$0. $0.$0. $0.$0.

Cj-ZjCj-Zj ---------- $6.$6. $15.$15.


ON A COLUMN BY COLUMN BASIS, SUBTRACT Zj FROM CjON A COLUMN BY COLUMN BASIS, SUBTRACT Zj FROM Cj


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 2424 11 11 11 00 00

$0.$0. SS22 160160 55 1010 00 11 00

$0.$0. SS33 1010 00 11 00 00 11

ZjZj $0.$0. $0.$0. $0.$0. $0.$0. $0.$0. $0.$0.

Cj-ZjCj-Zj ---------- $6.$6. $15.$15. $0.$0.



CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 2424 11 11 11 00 00

$0.$0. SS22 160160 55 1010 00 11 00

$0.$0. SS33 1010 00 11 00 00 11

ZjZj $0.$0. $0.$0. $0.$0. $0.$0. $0.$0. $0.$0.

Cj-ZjCj-Zj ---------- $6.$6. $15.$15. $0.$0. $0.$0.



CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 2424 11 11 11 00 00

$0.$0. SS22 160160 55 1010 00 11 00

$0.$0. SS33 1010 00 11 00 00 11

ZjZj $0.$0. $0.$0. $0.$0. $0.$0. $0.$0. $0.$0.

Cj-ZjCj-Zj ---------- $6.$6. $15.$15. $0.$0. $0.$0. $0.$0.



CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 2424 11 11 11 00 00

$0.$0. SS22 160160 55 1010 00 11 00

$0.$0. SS33 1010 00 11 00 00 11

ZjZj $0.$0. $0.$0. $0.$0. $0.$0. $0.$0. $0.$0.

Cj-ZjCj-Zj ---------- $6.$6. $15.$15. $0.$0. $0.$0. $0.$0.

THE COMPLETED Cj - Zj ROWTHE COMPLETED Cj - Zj ROW


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 2424 11 11 11 00 00

$0.$0. SS22 160160 55 1010 00 11 00

$0.$0. SS33 1010 00 11 00 00 11

ZjZj $0.$0. $0.$0. $0.$0. $0.$0. $0.$0. $0.$0.

Cj-ZjCj-Zj ---------- $6.$6. $15.$15. $0.$0. $0.$0. $0.$0.

Simplex Linear ProgrammingSimplex Linear Programming1st FEASIBLE SOLUTION

TRADITIONALLY THE FIRST FEASIBLE SOLUTIONPRODUCES NO PRODUCT AND HAS ALL

RESOURCES INTACT

XX11BLACK AND

WHITE TVs 00

XX22 COLOR TVs 0

SS11 CHASSIS 24

SS22 LABOR HOURS 160

SS33 COLOR TUBES 10


X1 ( B+W TVs ) = 0NOT IN THE BASISNOT IN THE BASIS

X2 ( Color TVs ) = 0NOT IN THE BASISNOT IN THE BASIS

Z ( Profit ) = $0.00SINCE XSINCE X11 AND X AND X2 2 = 0= 0

1st FEASIBLE SOLUTION

Toward 2nd Feasible SolutionToward 2nd Feasible Solution

CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 2424 11 11 11 00 00

$0.$0. SS22 160160 55 1010 00 11 00

$0.$0. SS33 1010 00 11 00 00 11

ZjZj $0.$0. $0.$0. $0.$0. $0.$0. $0.$0. $0.$0.

Cj-ZjCj-Zj ---------- $6.$6. $15.$15. $0.$0. $0.$0. $0.$0.

WE BRING THE MOST PROFITABLE TELEVISION ( X2 ) INTO THE SOLUTIONWE BRING THE MOST PROFITABLE TELEVISION ( X2 ) INTO THE SOLUTION

X2 BECOMES THE PIVOT COLUMNX2 BECOMES THE PIVOT COLUMN


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 2424 11 11 11 00 00

$0.$0. SS22 160160 55 1010 00 11 00

$0.$0. SS33 1010 00 11 00 00 11

ZjZj $0.$0. $0.$0. $0.$0. $0.$0. $0.$0. $0.$0.

Cj-ZjCj-Zj ---------- $6.$6. $15.$15. $0.$0. $0.$0. $0.$0.

WE COMPUTE EACH ROW’S RATIO OF QUANTITY DIVIDED BY ITS PIVOT COLUMN COEFFICIENTWE COMPUTE EACH ROW’S RATIO OF QUANTITY DIVIDED BY ITS PIVOT COLUMN COEFFICIENT

PIVOT COLUMNPIVOT COLUMN

2424

1616

1010


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 2424 11 11 11 00 00

$0.$0. SS22 160160 55 1010 00 11 00

$0.$0. SS33 1010 00 11 00 00 11

ZjZj $0.$0. $0.$0. $0.$0. $0.$0. $0.$0. $0.$0.

Cj-ZjCj-Zj ---------- $6.$6. $15.$15. $0.$0. $0.$0. $0.$0.

THE LOWEST POSITIVE RATIO DENOTES THE PIVOT ROWTHE LOWEST POSITIVE RATIO DENOTES THE PIVOT ROW


2424

1616

1010

PPIIVVOOTT

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MEANS WE CANONLY PRODUCETEN COLOR TVs

THIS IS THELIMITED RESOURCE


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 2424 11 11 11 00 00

$0.$0. SS22 160160 55 1010 00 11 00

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $0.$0. $0.$0. $0.$0. $0.$0. $0.$0. $0.$0.

Cj-ZjCj-Zj ---------- $6.$6. $15.$15. $0.$0. $0.$0. $0.$0.

REAL VARIABLE X2 REPLACES SLACK VARIABLE S3 IN THE BASISREAL VARIABLE X2 REPLACES SLACK VARIABLE S3 IN THE BASIS


PPIIVVOOTT

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TO PRODUCETEN COLOR TVs

MEANS THATALL COLOR

TUBES MUSTBE CONSUMED,MAKING S3 = 0,AND FORCING IT TO LEAVE THE BASIS


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 2424 11 11 11 00 00

$0.$0. SS22 160160 55 1010 00 11 00

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $0.$0. $0.$0. $0.$0. $0.$0. $0.$0. $0.$0.

Cj-ZjCj-Zj ---------- $6.$6. $15.$15. $0.$0. $0.$0. $0.$0.

THE PIVOT NUMBER IS LOCATED AT THE INTERSECTION OF THE PIVOT ROW AND PIVOT COLUMNTHE PIVOT NUMBER IS LOCATED AT THE INTERSECTION OF THE PIVOT ROW AND PIVOT COLUMN


PPIIVVOOTT

RROOWW


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 2424 11 11 11 00 00

$0.$0. SS22 160160 55 1010 00 11 00

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $0.$0. $0.$0. $0.$0. $0.$0. $0.$0. $0.$0.

Cj-ZjCj-Zj ---------- $6.$6. $15.$15. $0.$0. $0.$0. $0.$0.


PPIIVVOOTT

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THE PIVOTNUMBER

MUSTALWAYS

BE“1”

When the Pivot Number is Not “1”When the Pivot Number is Not “1”

CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.

BasisBasis QtyQty XX11 XX22 S1S1 S2 S3S3

$0.$0. 11

$0.$0. 1010

$15.$15. XX22 6060 1010 55 00 00 -10-10

ZjZj $0.$0. $0.$0. $0.$0. $0.$0. $0.$0. $0.$0.

Cj-ZjCj-Zj ---------- $15.$15.


PPIIVVOOTT

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EXAMPLEEXAMPLE

THE ENTIRE PIVOTROW MUST BE DIVIDEDBY WHATEVER NUMBER

NEEDED T0 FORCE AVALUE OF “1” FOR THE

PIVOT NUMBER

When the Pivot Number is Not “1”When the Pivot Number is Not “1”

CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. 11

$0.$0. 1010

$15.$15. XX22 1212 22 11 00 00 - 2- 2

ZjZj $0.$0. $0.$0. $0.$0. $0.$0. $0.$0. $0.$0.

Cj-ZjCj-Zj ---------- $15.$15.


PPIIVVOOTT

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EXAMPLEEXAMPLE

Toward 2Toward 2ndnd Feasible Solution Feasible SolutionROW TRANSFORMATIONROW TRANSFORMATION

Old 1st Row S1 24 1 1 1 0 0

Pivot Row S3 10 0 1 0 0 1

New 1st Row S1 14 1 0 1 0 -1

We subtract the pivot row from all other rows in the matrix ( except “Zj” and “Cj - Zj” ) in such a way as to force a “0” coefficient in the column above or below the pivot number.

SUBTRACT THE PIVOT ROW ( S3 ) FROM THE FIRST ROW ( S1 )

PivotNumber


Old 2nd Row S2 160 5 10 0 1 0

Pivot Row S3 10 0 1 0 0 1

New 2nd Row S2 0

SUBTRACT THE PIVOT ROW ( S3 ) FROM THE SECOND ROW ( S2 )

HERE, THE PIVOT ROW MUST BE MULTIPLIED BY “10” IN ORDER TO FORCE THE REQUIRED ZERO COEFFICIENT IN THE NEW ROW

PivotNumber


Old 2nd Row S2 160 5 10 0 1 0

Pivot Row S3 100 0 10 0 0 10

New 2nd Row S2 60 5 0 0 1 -10

SUBTRACT THE PIVOT ROW ( S3 ) FROM THE SECOND ROW ( S2 )

HERE, THE PIVOT ROW WAS MULTIPLIED BY “10” IN ORDER TO FORCE THE REQUIREDZERO COEFFICIENT IN THE NEW ROW

PivotNumber

Toward 2Toward 2ndnd Feasible Solution Feasible Solution

CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 1414 11 00 11 00 -1-1

$0.$0. SS22 6060 55 00 00 11 -10-10

$15.$15. XX22 1010 00 11 00 00 11

ZjZj

Cj-ZjCj-Zj

INSERTING THE TRANSFORMED ROWS INTO THE MATRIXINSERTING THE TRANSFORMED ROWS INTO THE MATRIX


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 1414 11 00 11 00 -1-1

$0.$0. SS22 6060 55 00 00 11 -10-10

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $150.$150.

Cj-ZjCj-Zj


$0.$0.

$0.$0.

$150.$150.

MULTIPLY EACH BASIS VARIABLE’S “Cj” BY ITS QUANTITY AND ADD THE RESULTSMULTIPLY EACH BASIS VARIABLE’S “Cj” BY ITS QUANTITY AND ADD THE RESULTS


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 1414 11 00 11 00 -1-1

$0.$0. SS22 6060 55 00 00 11 -10-10

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $150.$150. $0.$0.

Cj-ZjCj-Zj


$0.$0.

$0.$0.

$0.$0.

MULTIPLY EACH BASIS VARIABLE’S “Cj” BY ITS VARIABLE COEFFICIENTS AND ADD THE RESULTSMULTIPLY EACH BASIS VARIABLE’S “Cj” BY ITS VARIABLE COEFFICIENTS AND ADD THE RESULTS


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 1414 11 00 11 00 -1-1

$0.$0. SS22 6060 55 00 00 11 -10-10

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $150.$150. $0.$0. $15.$15.

Cj-ZjCj-Zj


$0.$0.

$0.$0.

$15.$15.



CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 1414 11 00 11 00 -1-1

$0.$0. S2S2 6060 55 00 00 11 -10-10

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $150.$150. $0.$0. $15.$15. $0.$0. $0.$0. $15.$15.

Cj-ZjCj-Zj


$0.$0.

$0.$0.


$0.$0.


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 1414 11 00 11 00 -1-1

$0.$0. SS22 6060 55 00 00 11 -10-10

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $150.$150. $0.$0. $15.$15. $0.$0. $0.$0. $15.$15.

Cj-ZjCj-Zj


$0.$0.

$0.$0.

$0.$0.



CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 1414 11 00 11 00 -1-1

$0.$0. SS22 6060 55 00 00 11 -10-10

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $150.$150. $0.$0. $15.$15. $0.$0. $0.$0. $15.$15.

Cj-ZjCj-Zj


$0.$0.

$0.$0.

$15.$15.



CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 1414 11 00 11 00 -1-1

$0.$0. SS22 6060 55 00 00 11 -10-10

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $150.$150. $0.$0. $15.$15. $0.$0. $0.$0. $15.$15.

Cj-ZjCj-Zj

THE COMPLETED Zj ROWTHE COMPLETED Zj ROW


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 1414 11 00 11 00 -1-1

$0.$0. SS22 6060 55 00 00 11 -10-10

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $150.$150. $0.$0. $15.$15. $0.$0. $0.$0. $15.$15.

Cj-ZjCj-Zj


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 1414 11 00 11 00 -1-1

$0.$0. SS22 6060 55 00 00 11 -10-10

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $150.$150. $0.$0. $15.$15. $0.$0. $0.$0. $15.$15.

Cj-ZjCj-Zj ------ $6.$6.

COMPUTING THE Cj - Zj ROWCOMPUTING THE Cj - Zj ROW



CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 1414 11 00 11 00 -1-1

$0.$0. SS22 6060 55 00 00 11 -10-10

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $150.$150. $0.$0. $15.$15. $0.$0. $0.$0. $15.$15.

Cj-ZjCj-Zj ------ $6.$6. $0.$0.




CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 1414 11 00 11 00 -1-1

$0.$0. SS22 6060 55 00 00 11 -10-10

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $150.$150. $0.$0. $15.$15. $0.$0. $0.$0. $15.$15.

Cj-ZjCj-Zj ------ $6.$6. $0.$0. $0.$0.



CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 1414 11 00 11 00 -1-1

$0.$0. SS22 6060 55 00 00 11 -10-10

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $150.$150. $0.$0. $15.$15. $0.$0. $0.$0. $15.$15.

Cj-ZjCj-Zj ------ $6.$6. $0.$0. $0.$0. $0.$0.



CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.$0.


$0.$0. SS11 1414 11 00 11 00 -1-1

$0.$0. SS22 6060 55 00 00 11 -10-10

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $150.$150. $0.$0. $15.$15. $0.$0. $0.$0. $15.$15.

Cj-ZjCj-Zj ------ $6.$6. $0.$0. $0.$0. $0.$0. -$15.-$15.


The 2The 2ndnd Feasible Solution Feasible Solution

CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 1414 11 00 11 00 -1-1

$0.$0. SS22 6060 55 00 00 11 -10-10

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $150.$150. $0.$0. $15.$15. $0.$0. $0.$0. $15.$15.

Cj-ZjCj-Zj ------ $6.$6. $0.$0. $0.$0. $0.$0. -$15.-$15.

THE COMPLETED Cj – Zj ROWTHE COMPLETED Cj – Zj ROW

The 2The 2ndnd Feasible Solution Feasible Solution

CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 1414 11 00 11 00 -1-1

$0.$0. SS22 6060 55 00 00 11 -10-10

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $150.$150. $0.$0. $15.$15. $0.$0. $0.$0. $15.$15.

Cj-ZjCj-Zj ------ $6.$6. $0.$0. $0.$0. $0.$0. -$15.-$15.

Simplex Linear ProgrammingSimplex Linear Programming22ndnd FEASIBLE SOLUTION FEASIBLE SOLUTION

XX11BLACK AND

WHITE TVs 0

XX22 COLOR TVs 10

SS11 CHASSIS 14

SS22 LABOR HOURSLABOR HOURS 60

SS33 COLOR TUBESCOLOR TUBES 0 TOTAL PROFIT = $150.00

Simplex Linear ProgrammingSimplex Linear Programming22ndnd FEASIBLE SOLUTION FEASIBLE SOLUTION

Z ( total profit ) = $150.00

X1 ( B + W TVs ) = 0NOT IN THE BASISNOT IN THE BASIS

S3 ( Color Tubes ) = 0NOT IN THE BASISNOT IN THE BASIS

Toward 3rd Feasible SolutionToward 3rd Feasible Solution

CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 1414 11 00 11 00 -1-1

$0.$0. SS22 6060 55 00 00 11 -10-10

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $150.$150. $0.$0. $15.$15. $0.$0. $0.$0. $15.$15.

Cj-ZjCj-Zj ------ $6.$6. $0.$0. $0.$0. $0.$0. -$15.-$15.

WE BRING BLACK + WHITE TELEVISIONS (X1) INTO THE SOLUTIONWE BRING BLACK + WHITE TELEVISIONS (X1) INTO THE SOLUTION

X1X1 BECOMES THE PIVOT COLUMNBECOMES THE PIVOT COLUMN


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 1414 11 00 11 00 -1-1

$0.$0. SS22 6060 55 00 00 11 -10-10

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $150.$150. $0.$0. $15.$15. $0.$0. $0.$0. $15.$15.

Cj-ZjCj-Zj ------ $6.$6. $0.$0. $0.$0. $0.$0. -$15.-$15.

WE COMPUTE EACH ROW’S RATIO OF QUANTITY DIVIDED BY ITS PIVOT COLUMN COEFFICIENTWE COMPUTE EACH ROW’S RATIO OF QUANTITY DIVIDED BY ITS PIVOT COLUMN COEFFICIENT

THE PIVOT COLUMNTHE PIVOT COLUMN

1414

1212

00


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 1414 11 00 11 00 -1-1

$0.$0. SS22 6060 55 00 00 11 -10-10

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $150.$150. $0.$0. $15.$15. $0.$0. $0.$0. $15.$15.

Cj-ZjCj-Zj ------ $6.$6. $0.$0. $0.$0. $0.$0. -$15.-$15.

THE LOWEST POSITIVE RATIO DENOTES THE PIVOT ROWTHE LOWEST POSITIVE RATIO DENOTES THE PIVOT ROW


1414

1212

00

PPIIVVOOT T

RROOWW

MEANS WE CANONLY PRODUCE

12 B+W TVsTHIS IS THE

LIMITED RESOURCE


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 1414 11 00 11 00 -1-1

$6.$6. XX11 6060 55 00 00 11 -10-10

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $150.$150. $0.$0. $15.$15. $0.$0. $0.$0. $15.$15.

Cj-ZjCj-Zj ------ $6.$6. $0.$0. $0.$0. $0.$0. -$15.-$15.

REAL VARIABLE X1 REPLACES SLACK VARIABLE S2 IN THE BASISREAL VARIABLE X1 REPLACES SLACK VARIABLE S2 IN THE BASIS


PPIIVVOOT T

RROOWW

TO PRODUCE12 B+W TVs

MEANS THATALL REMAINING LABOR HOURS

MUST BECONSUMED,

MAKING S2 = 0, AND FORCING IT TO LEAVE THE BASIS


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 1414 11 00 11 00 -1-1

$6.$6. XX11 6060 55 00 00 11 -10-10

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $150.$150. $0.$0. $15.$15. $0.$0. $0.$0. $15.$15.

Cj-ZjCj-Zj ------ $6.$6. $0.$0. $0.$0. $0.$0. -$15.-$15.

THE PIVOT NUMBER IS LOCATED AT THE INTERSECTION OF THE PIVOT ROW AND PIVOT COLUMNTHE PIVOT NUMBER IS LOCATED AT THE INTERSECTION OF THE PIVOT ROW AND PIVOT COLUMN


PPIIVVOOT T

RROOWW


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 1414 11 00 11 00 -1-1

$6.$6. XX11 6060 55 00 00 11 -10-10

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $150.$150. $0.$0. $15.$15. $0.$0. $0.$0. $15.$15.

Cj-ZjCj-Zj ------ $6.$6. $0.$0. $0.$0. $0.$0. -$15.-$15.THE PIVOT COLUMNTHE PIVOT COLUMN

PPIIVVOOT T

RROOWW

THE PIVOT

NUMBERMUST

ALWAYSBE“1”


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 1414 11 00 11 00 -1-1

$6.$6. XX11 1212 11 00 00 .2.2 -2.0-2.0

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $150.$150. $0.$0. $15.$15. $0.$0. $0.$0. $15.$15.

Cj-ZjCj-Zj ------ $6.$6. $0.$0. $0.$0. $0.$0. -$15.-$15.


PPIIVVOOT T

RROOWW

DIVIDETHE

ENTIREPIVOT ROW

BY“5”

Toward 3Toward 3rdrd Feasible Solution Feasible SolutionROW TRANSFORMATIONROW TRANSFORMATION

Old 1st Row S1 14 1 0 1 0 -1

Pivot Row X1 12 1 0 0 .2 - 2.0

New 1st Row S1 2 0 0 1 -.2 1

SUBTRACT THE PIVOT ROW ( X1 ) FROM THE FIRST ROW ( S1 )

HERE, THERE WAS NO NEED TO MULTIPLY THE PIVOT ROW BY ANY NUMBER IN ORDER TO FORCE A ZERO COEFFICIENT BELOW THE PIVOT NUMBER

PivotNumber

Toward 3Toward 3rdrd Feasible Solution Feasible SolutionROW TRANSFORMATIONROW TRANSFORMATION

Old 3rd Row X2 10 0 1 0 0 1

Pivot Row X1 12 1 0 0 .2 - 2.0

New 3rd Row X2 10 0 1 0 0 1

SUBTRACT THE PIVOT ROW ( X1 ) FROM THE THIRD ROW ( X2 )

HERE, THERE WAS NO NEED TO TRANSFORM ROW “3” BECAUSE IT ALREADY HAD A ZERO COEFFICIENT BELOW THE PIVOT NUMBER

PivotNumber


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 22 00 00 11 -.2-.2 11

$6.$6. XX11 1212 11 00 00 .2.2 -2.0-2.0

$15.$15. XX22 1010 00 11 00 00 11

ZjZj

Cj-ZjCj-Zj

INSERTING THE TRANSFORMED ROWS INTO THE MATRIXINSERTING THE TRANSFORMED ROWS INTO THE MATRIX


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 22 00 00 11 -.2-.2 11

$6.$6. XX11 1212 11 00 00 .2.2 -2.0-2.0

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $222.$222.

Cj-ZjCj-Zj


$0.$0.

$72.$72.

$150.$150.

MULTIPLY EACH BASIS VARIABLE’S “Cj” BY ITS QUANTITY AND ADD THE RESULTSMULTIPLY EACH BASIS VARIABLE’S “Cj” BY ITS QUANTITY AND ADD THE RESULTS


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 22 00 00 11 -.2-.2 11

$6.$6. XX11 1212 11 00 00 .2.2 -2.0-2.0

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $222.$222. $6.$6.

Cj-ZjCj-Zj


$0.$0.

$6.$6.

$0.$0.



CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. S1S1 22 00 00 11 -.2-.2 11

$6.$6. X1X1 1212 11 00 00 .2.2 -2.0-2.0

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $222.$222. $6.$6. $15.$15.

Cj-ZjCj-Zj


$0.$0.

$0.$0.

$15.$15.



CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 22 00 00 11 -.2-.2 11

$6.$6. XX11 1212 11 00 00 .2.2 -2.0-2.0

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $222.$222. $6.$6. $15.$15. $0.$0.

Cj-ZjCj-Zj


$0.$0.

$0.$0.

$0.$0.



CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 22 00 00 11 -.2-.2 11

$6.$6. XX11 1212 11 00 00 .2.2 -2.0-2.0

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $222.$222. $6.$6. $15.$15. $0.$0. $1.2$1.2

Cj-ZjCj-Zj


$0.$0.

$1.2$1.2

$0.$0.



CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 22 00 00 11 -.2-.2 11

$6.$6. XX11 1212 11 00 00 .2.2 -2.0-2.0

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $222.$222. $6.$6. $15.$15. $0.$0. $1.2$1.2 $3.0$3.0

Cj-ZjCj-Zj


$0.$0.

$15.$15.


-$12.-$12.


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 22 00 00 11 -.2-.2 11

$6.$6. XX11 1212 11 00 00 .2.2 -2.0-2.0

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $222.$222. $6.$6. $15.$15. $0.$0. $1.2$1.2 $3.0$3.0

Cj-ZjCj-Zj

THE COMPLETED Zj ROWTHE COMPLETED Zj ROW


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 22 00 00 11 -.2-.2 11

$6.$6. XX11 1212 11 00 00 .2.2 -2.0-2.0

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $222.$222. $6.$6. $15.$15. $0.$0. $1.2$1.2 $3.0$3.0

Cj-ZjCj-Zj


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 22 00 00 11 -.2-.2 11

$6.$6. XX11 1212 11 00 00 .2.2 -2.0-2.0

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $222.$222. $6.$6. $15.$15. $0.$0. $1.2$1.2 $3.0$3.0

Cj-ZjCj-Zj $0.$0.




CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 22 00 00 11 -.2-.2 11

$6.$6. XX11 1212 11 00 00 .2.2 -2.0-2.0

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $222.$222. $6.$6. $15.$15. $0.$0. $1.2$1.2 $3.0$3.0

Cj-ZjCj-Zj $0.$0. $0.$0.




CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 22 00 00 11 -.2-.2 11

$6.$6. XX11 1212 11 00 00 .2.2 -2.0-2.0

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $222.$222. $6.$6. $15.$15. $0.$0. $1.2$1.2 $3.0$3.0

Cj-ZjCj-Zj $0.$0. $0.$0. $0.$0.




CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 22 00 00 11 -.2-.2 11

$6.$6. XX11 1212 11 00 00 .2.2 -2.0-2.0

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $222.$222. $6.$6. $15.$15. $0.$0. $1.2$1.2 $3.0$3.0

Cj-ZjCj-Zj $0.$0. $0.$0. $0.$0. -$1.2-$1.2




CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 22 00 00 11 -.2-.2 11

$6.$6. XX11 1212 11 00 00 .2.2 -2.0-2.0

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $222.$222. $6.$6. $15.$15. $0.$0. $1.2$1.2 $3.0$3.0

Cj-ZjCj-Zj $0.$0. $0.$0. $0.$0. -$1.2-$1.2 -$3.0-$3.0



The 3rd Feasible SolutionThe 3rd Feasible Solution

CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 22 00 00 11 -.2-.2 11

$6.$6. XX11 1212 11 00 00 .2.2 -2.0-2.0

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $222.$222. $6.$6. $15.$15. $0.$0. $1.2$1.2 $3.0$3.0

Cj-ZjCj-Zj $0.$0. $0.$0. $0.$0. -$1.2-$1.2 -$3.0-$3.0

THE COMPLETED Cj – Zj ROWTHE COMPLETED Cj – Zj ROW


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 22 00 00 11 -.2-.2 11

$6.$6. XX11 1212 11 00 00 .2.2 -2.0-2.0

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $222.$222. $6.$6. $15.$15. $0.$0. $1.2$1.2 $3.0$3.0

Cj-ZjCj-Zj $0.$0. $0.$0. $0.$0. -$1.2-$1.2 -$3.0-$3.0


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 22 00 00 11 -.2-.2 11

$6.$6. XX11 1212 11 00 00 .2.2 -2.0-2.0

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $222.$222. $6.$6. $15.$15. $0.$0. $1.2$1.2 $3.0$3.0

Cj-ZjCj-Zj $0.$0. $0.$0. $0.$0. -$1.2-$1.2 -$3.0-$3.0

THERE ARE NO POSITIVE NUMBERS IN THE Cj – Zj ROWTHERE ARE NO POSITIVE NUMBERS IN THE Cj – Zj ROW

THETHEOPTIMALOPTIMAL

SOLUTIONSOLUTION

Simplex Linear ProgrammingSimplex Linear Programming33rdrd AND OPTIMAL FEASIBLE SOLUTION AND OPTIMAL FEASIBLE SOLUTION

X1 = 12X2 = 10

S1 = 2S2 = 0S3 = 0

Z = $222.00

Produce twelvetwelve black and whitetelevisionsProduce tenten color televisions

There are twotwo chassis left overThere are nono labor hours left overThere are nono color tubes left over

Total profit realized is $222.00$222.00

A RECORD SHOULD BE KEPT OF WHAT EACH VARIABLE REPRESENTS


CjCj $6.$6. $15.$15. $0.$0. $0.$0. $0.


$0.$0. SS11 22 00 00 11 -.2-.2 11

$6.$6. XX11 1212 11 00 00 .2.2 -2.0-2.0

$15.$15. XX22 1010 00 11 00 00 11

ZjZj $222.$222. $6.$6. $15.$15. $0.$0. $1.2$1.2 $3.0$3.0

Cj-ZjCj-Zj $0.$0. $0.$0. $0.$0. -$1.2-$1.2 -$3.0-$3.0

THE SLACK VARIABLE SHADOW PRICESTHE SLACK VARIABLE SHADOW PRICES

ALWAYS FOUND IN THE Cj – Zj ROWALWAYS FOUND IN THE Cj – Zj ROW

Simplex Linear ProgrammingSimplex Linear Programming33rdrd AND OPTIMAL FEASIBLE SOLUTION SHADOW PRICES AND OPTIMAL FEASIBLE SOLUTION SHADOW PRICES

S1 = $0.00

S2 = - $1.20

S3 = - $3.00

WE WOULD BE WILLING TO PAYWILLING TO PAYUP TO ZERO DOLLARSZERO DOLLARS FOR ADDITIONAL CHASSIS

WE WOULD BE WILLING TO PAYWOULD BE WILLING TO PAYUP TO $1.20$1.20 FOR ADDITIONALLABOR HOURS

WE WOULD BE WILLING TO PAYUP TO $3.00$3.00 FOR ADDITIONALCOLOR TUBES

THIS PARTICULAR SIMPLEX METHOD DISPLAYS NEGATIVE VALUES FOR POSITIVESHADOW PRICES. THIS IS CALLED THE “MIRROR EFFECT”.

Simplex Linear Programming under Simplex Linear Programming under QM for WINDOWSQM for WINDOWS

Template

Simplex Linear Simplex Linear ProgrammingProgramming

MaximizationMaximization

simplex linear programming i. concept ii. model template iii. class example iv. procedure v....

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labor hours0x1 1x2 1s3

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