EM 540 Operations Research

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• EM 540 Operations Research/
• DecS 581 Operations Management

Chapter 9 Linear Programming The Simplex Method
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Chapter 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
• 9.6 Review of Procedures for Solving LP
  Maximization Problems

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Chapter Outline - continued

• 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

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Learning Objectives

• Students will be able to
• Convert LP constraints to equalities with slack,
  surplus, and artificial variables.
• Set up and solve both simple maximization and
  minimization LP problems with simplex tableaus.
• Interpret the meaning of the numbers in a simplex
  tableau.

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Learning Objectives - continued

• 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.

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Flair Furniture Company
Hours Required to Produce One Unit
Available Hours This Week
X1 Tables
X2 Chairs
Department
Carpentry Painting/Varnishing
4 2
3 1
240 100
Profit/unit Constraints
7
5
Maximize
Objective
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George Dantzig studied mathematics at the
University of Maryland, receiving his A.B. in
1936. The following year he received an M.A. in
mathematics from the University of Michigan.
Dantzig worked as a Junior Statistician in the
U.S. Bureau of Labor Statistics from 1937 to
1939, then, from 1941 to 1946, he was head of the
Combat Analysis Branch, U.S.A.F. Headquarters
Statistical Control. He received his doctorate in
mathematics from the University of California,
Berkeley in 1946. In that year he was appointed
Mathematical Advisor for USAF Headquarters. In
1947 Dantzig made the contribution to mathematics
for which he is most famous, the simplex method
of optimization. It grew out of his work with the
U.S. Air Force where he become an expert on
planning methods solved with desk calculators. In
fact this was known as "programming", a military
term that, at that time, referred to plans or
schedules for training, logistical supply or
deployment of men.
Born 8 Nov 1914 in Portland, Oregon
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Dantzigs Challenge
However, writing in 1991, Dantzig noted that-
... it is interesting to note that the original
problem that started my research is still
outstanding - namely the problem of planning or
scheduling dynamically over time, particularly
planning dynamically under uncertainty. If such a
problem could be successfully solved it could
eventually through better planning contribute to
the well-being and stability of the world.
We solve this in EM 530 Applications in
Constraints Management
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Math Review
One Equation with One Unknown
Eq. 1 X5 Eq. 2 Y62-10/5
One Equation with Two Unknowns
Eq. 1 XY7 Eq. 2 Y is a function of X
Y(X)7-X
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More Math
Two Equations with Two Unknowns
Eq. 1 XY7 Eq. 2 X - Y1
Adding these two equations (equalities) together
Eq. 3 2X0Y8
Or
Eq. 4 X 4
Substituting into Eq. 1
Eq. 5 4 Y7Eq. 6 Y3
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Checking
We started with
Eq. 1 XY7 Eq. 2 X - Y1
We found
Eq. 4 X 4
Eq. 6 Y3
Checking
Eq. 1 4 37 Eq. 2 4 - 31
Looks Correct!
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Try another one a bit tougher
Lets try
Eq. 1 2X 1Y 100 Eq. 2 4X 3Y 240
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Solving for X
Lets try
Eq. 1 2X 1Y 100 Eq. 2 4X 3Y 240
Eq. 3 6X 3Y 300
Eq. 2 4X 3Y 240
Subtracting Eq. 2 from Eq. 3 gives
Eq. 4 2X 0Y 60
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Solving for Y
Then we have
Eq. 2 4X 3Y 240
Eq. 4 2X 0Y 60
Eq. 5 4X 0Y 120
Eq. 2 4X 3Y 240
Subtracting Eq. 5 from Eq. 2 gives
Eq. 6 0X 3Y 120
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We have X and Y
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You may remember Matrix Manipulation
We write
Eq. 1 2X 1Y 100 Eq. 2 4X 3Y 240
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Its Easy to Solve A Matrix
To solve this Matrix
X Y RHS Row 1
2 1 100 Row 2 4 3 240
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Its been a while, so just for fun!
We start with
2 1 100 4 3 240
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A bit more Manipulation
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And the Answer...
Yeah, no math error!
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But with InEqualities?
Lets try
Eq. 1 2X 1Y lt 100 Eq. 2 4X 3Y lt 240
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What is the answer?
Lets try
Eq. 1 2X 1Y lt 100 Eq. 2 4X 3Y lt 240
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We need to solve our InEqualities Differently
Cleverly, we can cause an inequality to vanish by
adding a dummy or Slack Variable to each
equation.
Eq. 1 2X 1Y lt 100 Eq. 2 4X 3Y lt 240
As long asS1 and S2 are gt 0 the original
inequalities hold.
New Eq. 1 2X 1Y S1 100 New Eq. 2
4X 3Y S2 240
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This may look messy, but a matrix solution
doesnt care
2X 1Y S1 100 4X 3Y S2
240
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So, Just for Practice (and to prepare us for the
Simplex Method)
Lets solve this inequality matrix
X Y S1 S2 RHS 2 1 1 0
100 4 3 0 1 240
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Solving is the same fashion as before
We start with
2 1 1 0 100 4 3 0 1 240
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Then
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Dividing Each Row
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Dividing Each Row
The Slack Variables (in conjunction with the
Objective Function (not shown or used here) would
tell us if X30 and Y40 was not the correct
solution!
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Now, lets revisit Flair Furniture
Let X1 Number of Tables produced Let X2
Number of Chairs produced Where profits are 7
per table and 5 per chair Maximize 7 X1
5X2 Subject to Paint 2X1 1X2
lt 100 Carpenter 4X1 3X2 lt240 Where
X1, X2 gt0
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Flair Furniture Company's Feasible Region
Corner Points
X2
100 80 60 40 20
B (0,80)
Number of Chairs
C (30,40)
Feasible Region
D (50,0)
0 20 40 60 80 100
X1
Number of Tables
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Flair Furniture - Adding Slack Variables
Constraints
240
3
4


)
(carpentry

X
X
2
1
100
1
2


)
varnishing



(painting

X
X
2
1
Constraints with Slack Variables
240
3
4



)
(carpentry



S
X
X
1
2
1
100
1
2



)
varnishing


(painting

S

X
X
2
2
1

Objective Function
5
7

X
X
2
1
Objective Function with Slack Variables
0
0
5
7



S
S
X
X
2
1
2
1
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Flair Furnitures Initial Simplex Tableau
Profit per Unit Column
Real Variables Columns
Production Mix Column
Constant Column
Solution Mix
X1
X2
S1
S2
2
1
1
0
S1
100
4
3
0
1
S2
240
?(Cj X1)
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Flair Furnitures Initial Simplex Tableau
Profit per Unit Column
Real Variables Columns
Slack Variables Columns
Production Mix Column
Constant Column
Profit per unit row
Cj
7
5
0
0
Solution Mix
X1
X2
S1
S2
Quantity
Constraint equation rows
2
1
1
0
0
S1
100
4
3
0
1
0
S2
240
Gross profit row
0
0
0
0
0
Zj
Net profit row
7
5
0
0
Cj - Zj
0
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Which Column Gives the Best Improvement?
Profit per Unit Column
Real Variables Columns
Slack Variables Columns
Production Mix Column
Constant Column
Profit per unit row
Cj
7
5
0
0
Solution Mix
X1
X2
S1
S2
Quantity
Constraint equation rows
2
1
1
0
0
S1
100
4
3
0
1
0
S2
240
Gross profit row
0
0
0
0
0
Zj
Net profit row
7
5
0
0
Cj - Zj
0
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How Far up the Curve can we go? (Least Positive)
Profit per Unit Column
Real Variables Columns
Slack Variables Columns
Production Mix Column
Constant Column
Profit per unit row
Cj
7
5
0
0
Solution Mix
X1
X2
S1
S2
Quantity
Constraint equation rows
100
2
1
1
0
0
S1
4
3
0
1
0
S2
240
Gross profit row
0
0
0
0
0
Zj
Net profit row
7
5
0
0
Cj - Zj
0
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Pivot Row, Pivot Number Identified in the Initial
Simplex Tableau
The math approach will be Drive the Pivot
Number to 1. Cause all other numbers in the
Pivot Column to go to Zero. Adjust Cj and sums.
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Calculating the New X1 Row
7
5
0
0
Solution Mix
X1
X2
S1
S2
Quantity
2
1
1
0
S1
100
40
Calculating the new S2 line
X1
X2
S1
S2
Quantity
1
1/2
1/2
0
X1
50
4
3
0
1
S2
240
Subtracting New X1 from S2
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After the First Simplex Tableau for Flair
Furniture
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Which Column has the Best Improvement Option?
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How far up the curve can we go? (Least Positive)
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Pivot Row, Column, and Number Identified in
Second Simplex Tableau
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The Pivot Number is Already 1
5 X2
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Adjusting the X1 Row
Pivot column
X1
X2
S1
S2
1
1/2
1/2
0
X1
50
0
1
-2
1
S2
40
Subtracting New S2 From X1
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Are There Further Improvements Possible?
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Final Simplex Tableau for the Flair Furniture
Problem
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Why is it Final?
We recognize the solution, but could it be
better? The Shadow prices are all zero or
negative. Cant get any better. So, its final.
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Total Value then is
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Simplex Steps for Maximization

• 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.

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Surplus Artificial Variables
Constraints
Constraints-Surplus Artificial Variables
Objective Function
Objective Function-Surplus Artificial Variables
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Simplex Steps for Minimization

• 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.

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Special CasesInfeasibility in Minimization
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Special Cases Unboundedness Maximize
Solution is Infinite
Pivot Column
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Special CasesDegeneracy
10/1/4 20/4 10/2
To Fix Remove Redundant Constraint
4 X1 0Y1 lt20 2 X1 0Y1 lt10are the same
line.
Can lock up computer iterations in loops
Pivot Column
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Special CasesMultiple Optima
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Sensitivity AnalysisHigh Note Sound Company
120
50


X


Max
X
1
2

to
Subject
80
4
2


X
X
2
1
60
1
3


X
X
2
1
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Sensitivity AnalysisHigh Note Sound Company
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Simplex SolutionHigh Note Sound Company
So, Whats the answer?
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Simplex SolutionHigh Note Sound Company
Discovering the Solution
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Non-basic Objective Function Coefficients
How much does the Cj for X1 have to increase
before it becomes Basic?
Cj -(Cj-Zj) is Break
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Basic Objective Function Coefficients
X2 wont leave until Cj for X2 drops 20.
?gt-20
?gt-120
S1 enter until Cj for X2 drops 120.
Sensitivity Analysis about the Solution Point!
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Simplex SolutionHigh Note Sound Company
Objective increases by 30 if 1 additional hour of
electricians time is available.
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Shadow Prices

• Shadow Price Value of One Additional Unit of a
  Scarce Resource
• Found in Final Simplex Tableau in Cj-Zj Row
• Negatives of Numbers in Slack Variable Column

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Steps 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.

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Primal Dual
The Dual is very helpful at speeding massive
solutions of certain types and provides some
additional information.
Dual



Fold over
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Comparison of the Primal and Dual Optimal Tableaus
The solution of the Dual is the Shadow price of
the Primal
Cj
50
120
0
0
Solution Mix
Quantity
X1
X2
S1
S2
Primals Optimal Solution
7
X2
20
1/2
1
1/4
0
5
S2
40
5/2
0
-1/4
1
Zj
2,400
60
120
30
0
Cj - Zj
-10
0
-30
0
Duals Optimal Solution
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Additional Pointers

• During your Homework, step through the Tableaus
  and watch as QM for Windows Calculates each
  cycle.

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(No Transcript)
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Cost Data at WXYZ
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Decision Options at WXYZ

• Accounting Produce items with Highest Profit
  Margin
• Z (52)(16 ea) - (20)(16) - (2 workers)(8
  hours)(10/hour) 352 /day
• Marketing Sell the product with the Highest
  Selling Price (commission)
• Y (55)(16 ea) - (25)(16) - (2 )(8 )(10)
  320 /day

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Other Opinions at WXYZ

• Production Produce items with Most Efficiency
  (Keep everyone busy)
• X (50)(24 ea) - (25)(24) - (2 workers)(8
  hours)(10/hour) 440 /day
• Global Thinking Produce items with Most
  Throughput per unit time of Constraint
• W (50)(24 ea) - (20)(24) - (2)(8)(10)
  560 /day

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PQ Corp
This example problem taken from Sifting
Infomation Out of the Data Ocean The Haystack
Syndrome by Eli Goldratt Used by permission,
James R. Holt, Associate Professor, Engineering
Management, Washington State University-Vancouver,
Jan 1999
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Marginal Profit LP of PQUsing Cost Accounting
Maximize 7.20 P 28.50Q s.t. A
15 P 10 Qlt2400 B
15 P 30 Qlt2400 C 15
P 5 Qlt2400 D 15 P
5 Qlt2400Demand P 1 P
lt100Demand Q
1 Qlt50
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LP PQ Solution
Will we actually make any money if we make 60 P
and 50 Q? We must pay the bills of 6000 per
week! Why wont we? Is there any other option?
Why is it better?
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PQ Corp
This example problem taken from Sifting
Infomation Out of the Data Ocean The Haystack
Syndrome by Eli Goldratt Used by permission,
James R. Holt, Associate Professor, Engineering
Management, Washington State University-Vancouver,
Jan 1999
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Marginal Profit LP of PQUsing Contribution Margin
Maximize 45 P 60 Q s.t. A
15 P 10 Qlt2400 B 15
P 30 Qlt2400 C 15 P
5 Qlt2400 D 15 P
5 Qlt2400Demand P 1 P
lt100Demand Q 1
Qlt50
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Revised PQ Problem
Will we actually make any money if we make 100 P
and 30 Q? We must pay the bills of 6000 per
week! Yes, we will generate 6300 this week!