TM 631 Optimization Fall 2006 Dr. Frank Joseph Matejcik

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TM 631 Optimization Fall 2006Dr. Frank Joseph
Matejcik
4th Session Ch. 4 Duality Theory 9/25/06
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Activities

• Review assignments and resources
• Assignment
• weird way of numbering problems 6.3-1, 6.3-5,
  and 6.8-3(abce)
• Chapter 6 H L (Limited discussion)

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Tentative Schedule
Chapters Assigned 8/28/2006 1,
2 ________ 9/04/2006 Holiday 9/11/2006 3
3.1-8,3.2-4,3.6-3 9/18/2006 4 4.3-6, 4.4-6,
4.7-6 9/25/2006 6 6.3-1, 6.3-5, and
6.8-3(abce) 10/02/2006 Exam 1 10/09/2006 Holiday 1
0/16/2006 8 10/23/2006 8 cont. 10/30/2006 21 11/06
/2006 Exam 2
Chapters Assigned 11/13/2006 9
11/20/2006 9 cont. 11/27/2006 11 12/04/2006 11
or 13 12/11/2005 Final
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Web Resources

• Class Web site on the HPCnet system
• http//sdmines.sdsmt.edu/sdsmt/directory/courses/2
  006fa/tm631021
• Streaming video http//its.sdsmt.edu/Distance/
• The same class session that is on the DVD is on
  the stream in lower quality. http//www.flashget.c
  om/ will allow you to capture the stream more
  readily and review the lecture, anywhere you can
  get your computer to run.
• Answers not posted, yet.

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6.1Alt Model K-Corp

• Max Z 3X1 5X2
• s.t.
• X1 lt 8
• X2 lt 6
• 3X1 4X2 lt 36
• X1 gt 0
• X2 gt 0
• where Z profit (in 1,000,000s)

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6.1Alt Primal / Dual

• Max Z 3X1 5X2 Min Yo 8Y1 6Y2 36Y3
• s.t. s.t.
• X1 lt 8 Y1 3Y3 gt 3
• X2 lt 6 Y2 4Y3 gt 5
• 3X1 4X2 lt 36 Y1, Y2 gt 0
• X1 , X2 gt 0

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6.1Alt Primal / Dual (General Case)
Primal Dual
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6.1Alt Primal / Dual (Relationship)
Like Table 6.2
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6.1Alt Primal / Dual (Relationship)
The only feasible solutions to primal are those
that satisfy optimality conditions for primal
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6.1Alt Primal / Dual (Relationship)
Primal works feasible to optimal Dual works
optimal to feasible
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6.1Alt Primal / Dual (Relationship)
Primal works feasible to optimal Dual works
optimal to feasible
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6.1Alt Primal / Dual Relationships

• Weak duality property, if x is a feasible
  solution for primal and y is feasible solution
  for dual,
• cx lt yb
• Strong duality property, if x is optimal for
  primal and y is optimal for dual,
• cx yb

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6.1 Primal / Dual Relationships

• Complementary Solutions, each iteration of
  simplex identifies a CPF solution x for primal
  and a complementary solution y for dual
• cx yb
• if x is not optimal for primal, then y is not
  feasible for dual

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6.1Alt Primal / Dual Relationships

• Complementary Slackness Property
• xj basic in primal ysi nonbasic in dual
• xj nonbasic in primal ysi basic in dual
• xsj basic in primal yi nonbasic in dual
• xsj basic in primal yi nonbasic in dual

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6.1Alt Primal / Dual Relationships

• Duality Theorem
• 1. If one problem has feasible solutions and a
• bounded objective function, then so does the
  other problem.
• 2. If one problem has feasible solutions and an
• unbounded objective function, the other has no
  feasible solutions.
• 3. If one problem has no feasible solutions, the
• other has either no feasible solutions or an
  unbounded objective function.

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6.3 Complementary Slackness

• Max Z 3X1 5X2 Min Yo 8Y1 6Y2 36Y3
• s.t. s.t.
• X1 lt 8 Y1 3Y3 gt 3
• X2 lt 6 Y2 4Y3 gt 5
• 3X1 4X2 lt 36

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6.3 Complementary Slackness
Iteration 0

• x1 0 ys1 -3 Z yo 0
• x2 0 ys2 -5
• xs1 8 y1 0
• xs2 6 y2 0
• xs3 36 y3 0

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6.3 Complementary Slackness

• Min Yo 8Y1 6Y2 36Y3 ys1 -3
• s.t. ys2 -5
• Y1 3Y3 gt 3 y1 0
• Y2 4Y3 gt 5 y2 0
• y3 0
• Yo 8(0) 6(0) 36(0) 0
• 0 3(0) gt 3
• 0 4(0) gt 5

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6.3 Complementary Slackness
Iteration 1

• x1 0 ys1 -3 Z yo 30
• x2 6 ys2 0
• xs1 8 y1 0
• xs2 0 y2 5
• xs3 12 y3 0

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6.3 Complementary Slackness

• Min Yo 8Y1 6Y2 36Y3 ys1 -3
• s.t. ys2 0
• Y1 3Y3 gt 3 y1 0
• Y2 4Y3 gt 5 y2 5
• y3 0
• Yo 8(0) 6(5) 36(0) 30
• 0 3(0) gt 3
• 5 4(0) gt 5

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6.3 Complementary Slackness

• Min Yo 8Y1 6Y2 36Y3 ys1 -3
• s.t. ys2 -5
• Y1 3Y3 gt 3 y1 0
• Y2 4Y3 gt 5 y2 0
• y3 0
• Yo 8(0) 6(0) 36(0) 0
• 0 3(0) gt 3
• 0 4(0) gt 5

Infeasible
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6.3 Complementary Slackness
Iteration 2

• x1 4 ys1 0 Z yo 42
• x2 6 ys2 0
• xs1 4 y1 0
• xs2 0 y2 1
• xs3 0 y3 1

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6.3 Complementary Slackness

• Min Yo 8Y1 6Y2 36Y3 ys1 0
• s.t. ys2 0
• Y1 3Y3 gt 3 y1 0
• Y2 4Y3 gt 5 y2 1
• y3 1
• Yo 8(0) 6(1) 36(1) 42
• 0 3(1) gt 3
• 5 4(1) gt 5

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6.3 Complementary Slackness

• Min Yo 8Y1 6Y2 36Y3 ys1 0
• s.t. ys2 0
• Y1 3Y3 gt 3 y1 0
• Y2 4Y3 gt 5 y2 1
• y3 1
• Yo 8(0) 6(1) 36(1) 42
• 0 3(1) gt 3
• 5 4(1) gt 5

Feasible
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6.6 Sensitivity Analysis

y 0 1 1
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6.6 Sensitivity Analysis

y 0 1 1
Suppose we consider changing the right hand side
of constraint 3 3X1 4X2 lt 36 48
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6.6 Sensitivity Analysis

• b Sb ?

-
8
4
3
6
1
3
48
0
/
(
)
/
(
)



0
1
6
0
6
(
)
-

0
4
3
6
1
3
48
8
/
(
)
/
(
)
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6.6 Sensitivity Analysis

• New Solution
• Xs1 0
• X2 6
• X3 8

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6.7 Sensitivity Analysis
Consider range on b3. 24 lt b3 lt 48 optimal
basis at intersection constraints 2 3
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6.7 Changes in cj Parameters
Z 3X1 5X2 Now, suppose c2 goes to ?.
(4,6)
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6.7 Changes in cj Parameters
Z 3X1 5X2 If, 4 lt c2 lt ?, Optimal
remains at (4,6) with Z 42
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6.7 Reflection on Sensitivity

• Recall, the graphical changes on the right hand
  side.

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6.7 Shadow Prices
Consider range on b1. 4 lt b1 lt ? optimal
basis at intersection constraints 2 3
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6.7 Shadow Prices
Consider range on b2.
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6.7 Shadow Prices
Consider range on b2. 3 lt b2 lt 9 optimal
basis at intersection constraints 2 3
X2 lt 6
(8,3)
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6.7 Shadow Prices
Consider range on b3. 24 lt b3 lt 48 optimal
basis at intersection constraints 2 3
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6.7 Sensitivity Summary

• Max Z 3X1 5X2
• s.t.
• X1 lt 8 4 lt b1 lt ? y10
• X2 lt 6 3 lt b2 lt 9 y21
• 3X1 4X2 lt 36 24 lt b3 lt 48 y31
• X1, X2 gt 0

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6.7 Using Duality

• Recall,
• b Sb ?
• For, K-Corp

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6.7 Using Duality for Sensitivity

• b

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6.7 Using Duality for Sensitivity

• For basis to remain in solution,
• b
  gt 0

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6.7 Using Duality for Sensitivity

• Range on
• b
  gt 0

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6.7 Using Duality for Sensitivity

• Range on
• b
  gt 0

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6.7 Using Duality for Sensitivity

• Range on
• b
• 4 gt 0
• gt -4

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6.7 Using Duality for Sensitivity

• Range on
• b
• 4 gt 0
• gt -4 b1 gt 4

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6.7 Using Duality for Sensitivity

• Range on
• b
  gt 0

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6.7 Using Duality for Sensitivity

• Range on
• b

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6.7 Using Duality for Sensitivity

• Range on
• b

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6.7 Using Duality for Sensitivity

• Range on
• b

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6.7 Using Duality for Sensitivity

• Range on
• b
  gt 0

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6.7 Using Duality for Sensitivity

• Range on
• b

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6.7 Using Duality for Sensitivity

• Range on
• b

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6.7 Using Duality for Sensitivity
Summary b
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6.7 Sensitivity Summary

• Max Z 3X1 5X2
• s.t.
• X1 lt 8 4 lt b1 lt ? y10
• X2 lt 6 3 lt b2 lt 9 y21
• 3X1 4X2 lt 36 24 lt b3 lt 48 y31
• X1, X2 gt 0

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6.7 Changes to Objective Function

• Recall from graphical technique, changes to the
  objective function

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6.7 Changes in cj Parameters
Z 3X1 5X2 42
(4,6)
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6.7 Changes in cj Parameters
Z 3X1 5X2 Suppose c1 drops to 0.
(4,6)
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6.7 Changes in cj Parameters
Z 3X1 5X2 Now, suppose c1 goes to 3 3/4.
(4,6)
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6.7 Changes in cj Parameters
Z 3X1 5X2 If, 0 lt c1 lt 33/4, Optimal
remains at (4,6) with Z 42
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6.7 Changes in cj Parameters
Z 3X1 5X2 Suppose c2 drops to 4.
(4,6)
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6.7 Chages in cj Parameters
Z 3X1 5X2 Now, suppose c2 goes to ?.
(4,6)
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6.7 Changes in cj Parameters
Z 3X1 5X2 If, 4 lt c2 lt ?, Optimal
remains at (4,6) with Z 42
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6.7 Changes in cj Parameters
0 lt c1 lt 3 3/4 4 lt c2 lt ?
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6.7 Changes to Objective Function

y 0 1 1
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6.7 Changes to Objective Function

y 0 1 1
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6.7 Changes to Objective Function
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6.7 Changes to Objective Function
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6.7 Changes to Objective Function
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6.7 Changes to Objective Function
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6.7 Changes to Objective Function
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6.7 Changes to Objective Function
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6.7 Changes to Objective Function
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6.7 Changes to Objective Function
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6.7 Changes to Objective Function

³
D
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6.7 Changes to Objective Function

³
D
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6.7 Changes to Objective Function

³
D
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6.7 Sensitivity Summary
0 lt c1 lt 3 3/4, 4 lt c2 lt ?

• Max Z 3X1 5X2
• s.t.
• X1 lt 8 4 lt b1 lt 12 y10
• X2 lt 6 4 lt b2 lt 9 y21
• 3X1 4X2lt 36 24 lt b3 lt 48 y31
• X1, X2 gt 0

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6.7 Changes to Objective Function

y 0 1 1
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6.8 Sensitivity Analysis on a Spreadsheet

• Main Methods
• Make individual changes and resolve it.
• Generate a table of changes. The books method
  involves a specialty add-in called Solver Table.
• Get the sensitivity report. Chapter 4 describes
  the method of getting it.
• IOR Tutor also has
• Graphical Method and Sensitivity Analysis.