To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-1 © 2003 by Prentice...
-
Upload
bryana-danford -
Category
Documents
-
view
219 -
download
0
Transcript of To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-1 © 2003 by Prentice...
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-1 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Chapter 9Chapter 9
Linear Linear Programming:Programming: The Simplex The Simplex
MethodMethod
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-2 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Learning ObjectivesLearning Objectives
Students will be able to
• Convert LP constraints to
equalities with slack, surplus, and
artificial variables.
• Set up and solve both
maximization and minimization
LP problems with simplex
tableaus.
• Interpret the meaning of every
number in a simplex tableau.
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-3 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Learning Objectives - Learning Objectives - continuedcontinued
Students will be able to
• Recognize cases of infeasibility,
unboundedness, degeneracy, and
multiple optimal solutions in a
simplex output.
• Understand the relationship
between the primal and dual and
when to formulate and use the
dual.
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-4 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Chapter OutlineChapter Outline
9.1 Introduction
9.2 How to Set Up the Initial
Solution
9.3 Simplex Solution Procedures
9.4 The Second Simplex Tableau
9.5 Developing the Third
Simplex Tableau
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-5 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Chapter Outline - Chapter Outline - continuedcontinued
9.6 Review of Procedures for Solving LP Maximization Problems
9.7 Surplus and Artificial Variables
9.8 Solving Minimization Problems
9.9 Review of Procedures for Solving LP Minimization Problems
9.10 Special Cases
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-6 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Flair Furniture Flair Furniture CompanyCompany
Maximize:Objective: XX
Hours Required to Produce One Unit
DepartmentX1
TablesX2
Chairs
AvailableHours This
Week
CarpentryPainting/Varnishing
42
31
240100
Profit/unit
Constraints:
$7 $5
)varnishing & (painting
XX
)(carpentry XX
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-7 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Flair Furniture Company's Flair Furniture Company's
Feasible Region & Corner Feasible Region & Corner PointsPoints
Num
ber
of C
hair
s
100
80
60
40
20
0 20 40 60 80 100 X
X2
Number of Tables
B = (0,80)
C = (30,40)
D = (50,0)
FeasibleRegion
XX
XX
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-8 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Flair Furniture - Flair Furniture - Adding Slack Adding Slack
VariablesVariables
)varnishing & (painting XX
)(carpentry XX
Constraints:
Constraints with Slack Variables
)varnishing& (painting
)(carpentry
S XX
SXX
XX Objective Function
Objective Function with Slack Variables
SSXX
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-9 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Flair Furniture’s Initial Flair Furniture’s Initial Simplex TableauSimplex Tableau
ProfitperUnit
ColumnProd.Mix
Column
Real VariablesColumns Slack
VariablesColumns
ConstantColumn
Cj
SolutionMix X1 X2 S1 S2 Quantity
$7 $5 $0 $0Profit per
unit row
2 1 1 0
4 3 0 1
$0 $0 $0 $0
$7 $5 $0 $0
$0
$0
S1
S2
Zj
Cj - Zj
100
240
$0
$0
Constraintequation
rows
GrossProfit rowNet
Profit row
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-10 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Pivot Row, Pivot Number Pivot Row, Pivot Number Identified in the Initial Identified in the Initial
Simplex TableauSimplex Tableau
Cj
SolutionMix X1 X2 S1 S2 Quantity
$7 $5 $0 $0
2 1 1 0
4 3 0 1
$0 $0 $0 $0
$7 $5 $0 $0
$0
$0
S1
S2
Zj
Cj - Zj
100
240
$0
$0
Pivot row
Pivot number
Pivot column
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-11 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Completed Second Simplex Completed Second Simplex Tableau for Flair FurnitureTableau for Flair Furniture
Cj
SolutionMix X1 X2 S1 S2 Quantity
$7 $5 $0 $0
1 1/2 1/2 0
0 1 -2 1
$7 $7/2 $7/2 $0
$0 $3/2 -$7/2 $0
$7
$0
X1
S2
Zj
Cj - Zj
50
40
$350
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-12 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Pivot Row, Column, and Pivot Row, Column, and Number Identified in Second Number Identified in Second
Simplex TableauSimplex Tableau
Cj
SolutionMix X1 X2 S1 S2 Quantity
$7 $5 $0 $0
1 1/2 1/2 0
0 1 -2 1
$7 $7/2 $7/2 $0
$0 $3/2 -$7/2 $0
$7
$0
X1
S2
Zj
Cj - Zj
50
40
$350(TotalProfit)
Pivot row
Pivot number
Pivot column
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-13 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Calculating the New Calculating the New XX11 Row for Flair’s Third Row for Flair’s Third
TableauTableau
= - x10
3/2-1/230
11/21/20
50
(1/2)(1/2)(1/2)(1/2)(1/2)
(0)(1)(-2)(1)
(40)
= - x
= - x
= - x
= - x
row X
newin
number
ingCorrespond
number
pivot
above
Number
rowX
oldin
Number
RowX
Newin
Number
i i
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-14 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Final Simplex Tableau for Final Simplex Tableau for
the Flair Furniture the Flair Furniture ProblemProblem
Cj
SolutionMix X1 X2 S1 S2 Quantity
$7 $5 $0 $0
1 0 3/2 -1/2
0 1 -2 1
$7 5 $1/2 $3/2
$0 $0 -$1/2-$3/2
$7
$5
X1
X2
Zj
Cj - Zj
30
40
$410
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-15 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Simplex Steps for Simplex Steps for MaximizationMaximization
1. Choose the variable with the greatest positive Cj - Zj to enter the solution.
2. Determine the row to be replaced by selecting 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 greater than zero, return to Step 1.
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-16 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Surplus & Artificial Surplus & Artificial VariablesVariables
Constraints
Constraints-Surplus & Artificial Variables
XX
XXX
AXX
ASXXX
Objective Function
XXX :Min
MAMASXXX :Min
Objective Function-Surplus & Artificial Variables
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-17 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Simplex Steps for Simplex Steps for MinimizationMinimization
1. Choose the variable with the greatest
negative Cj - Zj to enter the solution.
2. Determine the row to be replaced by selecting 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.
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-18 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Special CasesSpecial CasesInfeasibilityInfeasibility
02M+21
M-
31
200Cj - Zj
1800+2M
0-21-M
31-
M
-285Zj
201-1-1000A2M
1000-12110X28
2000-13-201X15
QtyA2A1S2S1X2X1Sol
Mix
MM0085Cj
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-19 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Special Cases Special 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
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-20 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Special CasesSpecial CasesDegeneracyDegeneracy
Pivot Column
C j 5 8 2 0 0 0
SolutionMix
X1 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-Z j 3 0 6 16 0 0
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-21 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Special CasesSpecial CasesMultiple OptimaMultiple Optima
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
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-22 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Sensitivity AnalysisSensitivity AnalysisHigh Note Sound CompanyHigh Note Sound Company
XX
XX
X X
:toSubject
:Max
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-23 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Sensitivity AnalysisSensitivity AnalysisHigh Note Sound CompanyHigh Note Sound Company
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-24 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Simplex SolutionSimplex SolutionHigh Note Sound CompanyHigh Note Sound Company
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
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-25 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Nonbasic Objective Nonbasic 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
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-26 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Basic Objective Function Basic 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
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-27 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Simplex SolutionSimplex SolutionHigh Note Sound CompanyHigh Note Sound Company
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 ½ 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
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-28 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
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
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-29 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
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 primal
constraints 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 constraint coefficients.
• Constraint inequality signs are reversed.
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-30 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Primal & DualPrimal & Dual
Primal: Dual
XX
XX
Subject to:
UU
UU
Subject to:
XX :Max UU :Min
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
9-31 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Comparison of the Primal Comparison of the Primal and Dual Optimal Tableausand Dual Optimal Tableaus
Pri
mal
’s O
ptim
al S
olut
ion
Du
al’s
Opt
imal
Sol
utio
n
Cj
SolutionMix
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
Cj
SolutionMix
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