© copyright andrew hall, 2002 fomgt 353 introduction to management science lecture 18 slide 1...
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© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 1
Network Models
Lecture 18The Transportation Algorithm
II
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 2
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
3
Dummy
10000
0
0
1 2
New York Montreal
01
2
3
0
Worcester15000 10000
Boston0 10000
Hartford10000
The Initialized TransportationSimplex Tableau
• Recall at the end of the last lecture we had initialized the Transportation Simplex Tableau starting from the results of the Least Cost Starting Procedure.
LCSP
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 3
Transportation Simplex cont…
Identify Entering Variable
Transportation Simplex Pivot
Exists?
Yes
No
Identify New Basic Variables
Stop.
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 4
The Transportation Simplex Method – Identify Entering
Variable• Recall from the Generalized Simplex Method
that a variable is an Entering Variable in a Minimization problem if its Cij-Zij is the most negative of the Cij-Zij values of the non-basic variables.
• So we need to calculate Cij-Zij values for all the non-basic variables.
• So first we need to calculate Ui and Vj values for the Basic Variables using the relationship: Cij – Ui + Vj = 0
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 5
Transportation Simplex cont…
Identify the Entering Variable
Use The Basic Variables and The Relationship Cij – Ui + Vj = 0 to Calc a Ui or a Vj.
UnknownUi or Vj?
Use The Relationship Cij – Zij = Cij – Ui + Vj to Calc a (Cij-Zij) Value for one of the NonBasic Variables
Cij-ZijLeft toCalc?
Yes
Yes
No
No
All+ve?
Stop. Yes
Identify the Basic Variables
Pivot
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 6
The Transportation Simplex Method – Identify Entering
Variable
• We use the relationship Cij – Ui + Vj = 0 for x12 to calculate V2.
• C12 – U1 + V2 = 0 => 12 – 0 + V2 = 0 • => V2 = -12
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
3
Dummy
10000
0
0
1 2
New York Montreal
01
2
3
0
Worcester15000 10000
Boston0 10000
Hartford10000
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
3
Dummy
10000
0
0
1 2
New York Montreal
-12
01
2
3
0
Worcester15000 10000
Boston0 10000
Hartford10000
LCSP
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 7
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
3
Dummy
10000
0
0
1 2
New York Montreal
-12
01
2
3
0
Worcester15000 10000
Boston0 10000
Hartford10000
The Transportation Simplex Method – Identify Entering
Variable
• We use the relationship Cij – Ui + Vj = 0 for x13 to calculate V3.
• C13 – U1 + V3 = 0 => 100 – 0 + V3 = 0 • => V3 = -100
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
-100
3
Dummy
10000
0
0
1 2
New York Montreal
-12
01
2
3
0
Worcester15000 10000
Boston0 10000
Hartford10000
LCSP
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 8
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
-100
3
Dummy
10000
0
0
1 2
New York Montreal
-12
01
2
3
0
Worcester15000 10000
Boston0 10000
Hartford10000
The Transportation Simplex Method – Identify Entering
Variable
• We use the relationship Cij – Ui + Vj = 0 for x32 to calculate U3.
• C32 – U3 + V2 = 0 => 10 – U3 + (-12) = 0
• => U3 = -2
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
-100
3
Dummy
10000
0
0
1 2
New York Montreal
-12
0
-2
1
2
3
0
Worcester15000 10000
Boston0 10000
Hartford10000
LCSP
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 9
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
-100
3
Dummy
10000
0
0
1 2
New York Montreal
-12
01
2
3
0
Worcester15000 10000
Boston0 10000
Hartford10000
The Transportation Simplex Method – Identify Entering
Variable
• We use the relationship Cij – Ui + Vj = 0 for x31 to calculate V1.
• C31 – U3 + V1 = 0 => 5 – (-2) + V1 = 0 • => V1 = -7
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
-100
3
Dummy
10000
0
0
1 2
New York Montreal
-12
0
-2
1
2
3
0
Worcester15000 10000
Boston0 10000
Hartford10000
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
-100
3
Dummy
10000
0
0
1 2
New York Montreal
-7 -12
0
-2
1
2
3
0
Worcester15000 10000
Boston0 10000
Hartford10000
LCSP
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 10
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
-100
3
Dummy
10000
0
0
1 2
New York Montreal
-7 -12
0
-2
1
2
3
0
Worcester15000 10000
Boston0 10000
Hartford10000
The Transportation Simplex Method – Identify Entering
Variable
• We use the relationship Cij – Ui + Vj = 0 for x21 to calculate U2.
• C21 – U2 + V1 = 0 => 4 – U2 + (-7) = 0 • => -3 – U2 = 0 => -3 = U2
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
-100
3
Dummy
10000
0
0
1 2
New York Montreal
-7 -12
0
-3
-2
1
2
3
0
Worcester15000 10000
Boston0 10000
Hartford10000
LCSP
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 11
The Next Steps..
Identify the Entering Variable
Use The Basic Variables and The Relationship Cij – Ui + Vj = 0 to Calc a Ui or a Vj.
UnknownUi or Vj?
Use The Relationship Cij – Zij = Cij – Ui + Vj to Calc a (Cij-Zij) Value for one of the NonBasic Variables
Cij-ZijLeft toCalc?
Yes
Yes
No
No
All+ve?
Stop. Yes
Identify the Basic Variables
Pivot
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 12
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
-100
3
Dummy
10000
0
0
1 2
New York Montreal
-7 -12
0
-3
-2
1
2
3
0
Worcester15000 10000
Boston0 10000
Hartford10000
Cij - Zij
x11 -1
x22
x23
x33
• We use the relationship Cij – Zij = Cij – Ui + Vj for x11 to calculate C11 – Z11.
• We use the relationship Cij – Zij = Cij – Ui + Vj for x22 to calculate C22 – Z22.
• C22 – Z22 = C22 – U2 + V2 • C22 – Z22 = 11 – (-3) + (-12) = 2
The Transportation Simplex Method – Identify Entering
Variable
• C11 – Z11 = C11 – U1 + V1 • C11 – Z11 = 6 – 0 + (-7) = -1• We use the relationship Cij – Zij = Cij – Ui + Vj
for x33 to calculate C33 – Z33.• We use the relationship Cij – Zij = Cij – Ui + Vj
for x23 to calculate C23 – Z23.• C23 – Z23 = C23 – U2 + V3 • C23 – Z23 = 100 – (-3) + (-100) = 3• C33 – Z33 = C33 – U3 + V3 • C33 – Z33 = 100 – (-2) + (-100) = 2
Cij - Zij
x11 -1
x22 2
x23
x33
Cij - Zij
x11 -1
x22 2
x23 3
x33
Cij - Zij
x11 -1
x22 2
x23 3
x33 2
The Most –ve Cij – Zij is “–1” so x11 is
the Entering Variable!
LCSP
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 13
The Pivot
Identify the Entering Variable
Use The Basic Variables and The Relationship Cij – Ui + Vj = 0 to Calc a Ui or a Vj.
UnknownUi or Vj?
Use The Relationship Cij – Zij = Cij – Ui + Vj to Calc a (Cij-Zij) Value for one of the NonBasic Variables
Cij-ZijLeft toCalc?
Yes
Yes
No
No
All+ve?
Stop. Yes
Identify the Basic Variables
Pivot
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 14
• Label the Entering Variable with a “+” and the other variables around the Cycle “-”, “+” and “-”.
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
-100
3
Dummy
10000
0
0
1 2
New York Montreal
-7 -12
0
-3
-2
1
2
3
0
Worcester15000 10000
Boston0 10000
Hartford10000
• We look for a “Cycle” of at least 3 Basic Variables and the Entering Variable.
The Transportation Simplex Pivot
Cij - Zij
x11 -1
x22 2
x23 3
x33 2
x11 is the
Entering Variable!
+-
+ -
LCSP
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 15
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
-100
3
Dummy
10000
0
0
1 2
New York Montreal
-7 -12
0
-3
-2
1
2
3
0
Worcester15000 10000
Boston0 10000
Hartford10000
The Transportation Simplex Pivot
• Then work out the max x11 can be increased by without decreasing x12 or x31 below 0. i.e. Min(x12, x31) = 10000
+-
+ -
LCSP
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 16
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
-100
3
Dummy
10000
0
0
1 2
New York Montreal
-7 -12
0
-3
-2
1
2
3
0
Worcester15000 10000
Boston0 10000
Hartford10000
The Transportation Simplex Pivot
• Increase x11 and x32 by 10,000.• Decrease x12 and x31 by 10,000.
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
-100
3
Dummy
10000
0
0
1 2
New York Montreal
-7 -12
0
-3
-2
1
2
3
0
Worcester15000 20000
Boston10000 10000
Hartford10000
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
-100
3
Dummy
10000
0
0
1 2
New York Montreal
-7 -12
0
-3
-2
1
2
3
0
Worcester5000 20000
Boston10000 0
Hartford10000
LCSP
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 17
A Reminder…
Identify the Entering Variable
Use The Basic Variables and The Relationship Cij – Ui + Vj = 0 to Calc a Ui or a Vj.
UnknownUi or Vj?
Use The Relationship Cij – Zij = Cij – Ui + Vj to Calc a (Cij-Zij) Value for one of the NonBasic Variables
Cij-ZijLeft toCalc?
Yes
Yes
No
No
All+ve?
Stop. Yes
Identify the Basic Variables
Pivot
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 18
The Transportation Simplex
• Now, we start again calculating the revised values for Ui and Vj.
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
0
0
3
DummyNew York Montreal
1 2
1 Boston10000 0
0
2 Hartford10000 0
3 Worcester5000 20000
10000
LCSP
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 19
The Transportation Simplex
• We use the relationship Cij – Ui + Vj = 0 for x11 to calculate V1.
• C11 – U1 + V1 = 0 => 6 – 0 + V1 = 0 • => V1 = -6
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
0
0
3
DummyNew York Montreal
1 2
1 Boston10000 0
0
2 Hartford10000 0
3 Worcester5000 20000
10000
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
0
0
3
Dummy
-6
New York Montreal
1 2
1 Boston10000 0
0
2 Hartford10000 0
3 Worcester5000 20000
10000
LCSP
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 20
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
0
0
3
Dummy
-6
New York Montreal
1 2
1 Boston10000 0
0
2 Hartford10000 0
3 Worcester5000 20000
10000
The Transportation Simplex
• We use the relationship Cij – Ui + Vj = 0 for x13 to calculate V3.
• C13 – U1 + V3 = 0 => 100 – 0 + V3 = 0 • => V3 = -100
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
0
0
3
-100
Dummy
-6
New York Montreal
1 2
1 Boston10000 0
0
2 Hartford10000 0
3 Worcester5000 20000
10000
LCSP
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 21
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
0
0
3
-100
Dummy
-6
New York Montreal
1 2
1 Boston10000 0
0
2 Hartford10000 0
3 Worcester5000 20000
10000
The Transportation Simplex
• We use the relationship Cij – Ui + Vj = 0 for x21 to calculate U2.
• C21 – U2 + V1 = 0 => 4 – U2 + (-6) = 0 • => -2 – U2 = 0 => -2 = U2
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
0
0
3
-100
Dummy
-6
New York Montreal
1 2
1 Boston10000 0
0
2 Hartford10000 0
-2
3 Worcester5000 20000
10000
LCSP
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 22
The Transportation Simplex
• We use the relationship Cij – Ui + Vj = 0 for x31 to calculate U3.
• C31 – U3 + V1 = 0 => 5 – U3 + (-6) = 0 • => U3 = -1
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
0
0
3
-100
Dummy
-6
New York Montreal
1 2
1 Boston10000 0
0
2 Hartford10000 0
-2
3 Worcester5000 20000
10000
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
0
0
3
-100
Dummy
-6
New York Montreal
1 2
1 Boston10000 0
0
2 Hartford10000 0
-2
3 -1 Worcester5000 20000
10000
LCSP
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 23
The Transportation Simplex
• We use the relationship Cij – Ui + Vj = 0 for x32 to calculate V2.
• C32 – U3 + V2 = 0 => 10 – (-1) + V2 = 0 • => 11 + V2 = 0 => V2 = -11
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
0
0
3
-100
Dummy
-6
New York Montreal
1 2
1 Boston10000 0
0
2 Hartford10000 0
-2
3 -1 Worcester5000 20000
10000
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
0
0
3
-100
Dummy
-6 -11
New York Montreal
1 2
1 Boston10000 0
0
2 Hartford10000 0
-2
3 -1 Worcester5000 20000
10000
LCSP
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 24
Another Reminder…
Identify the Entering Variable
Use The Basic Variables and The Relationship Cij – Ui + Vj = 0 to Calc a Ui or a Vj.
UnknownUi or Vj?
Use The Relationship Cij – Zij = Cij – Ui + Vj to Calc a (Cij-Zij) Value for one of the NonBasic Variables
Cij-ZijLeft toCalc?
Yes
Yes
No
No
All+ve?
Stop. Yes
Identify the Basic Variables
Pivot
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 25
• C22 – Z22 = C22 – U2 + V2 • C22 – Z22 = 11 – (-2) + (-11) = 2• We use the relationship Cij – Zij = Cij – Ui + Vj
for x22 to calculate C22 – Z22.• We use the relationship Cij – Zij = Cij – Ui + Vj for
x12 to calculate C12 – Z12.• C12 – Z12 = C12 – U1 + V2 • C12 – Z12 = 12 – 0 + (-11) = 1
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
0
0
3
-100
Dummy
-6 -11
New York Montreal
1 2
1 Boston10000 0
0
2 Hartford10000 0
-2
3 -1 Worcester5000 20000
10000
The Transportation Simplex
• We use the relationship Cij – Zij = Cij – Ui + Vj for x33 to calculate C33 – Z33.
• We use the relationship Cij – Zij = Cij – Ui + Vj for x23 to calculate C23 – Z23.
• C23 – Z23 = C23 – U2 + V3 • C23 – Z23 = 100 – (-2) + (-100) = 2• C33 – Z33 = C33 – U3 + V3 • C33 – Z33 = 100 – (-1) + (-100) = 1
LCSP
Cij - Zij
x12 1
x22 2
x23 2
x33 1
Cij - Zij
x12 1
x22 2
x23 2
x33 1
Cij - Zij
x12 1
x22 2
x23 2
x33 1
Cij - Zij
x12 1
x22 2
x23 2
x33 1
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 26
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
0
0
3
-100
Dummy
-6 -11
New York Montreal
1 2
1 Boston10000 0
0
2 Hartford10000 0
-2
3 -1 Worcester5000 20000
10000
The Transportation Simplex
• There are no negative Cij – Zij values so stop. This is an optimal solution!
• We ship 10,000 from Boston to New York, with 10,000 slack capacity in Boston. We ship 10,000 from Hartford to New York, 5,000 from Worcester to New York and 20,000 from Worcester to Montreal.
Cij - Zij
x12 1
x22 2
x23 2
x33 1
LCSP
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 27
Another Example
Starting from the Initial Feasible Solution generated
using the Vogel’s Approximation Starting
Procedure
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 28
The Transportation Simplex Method – Identify Entering
Variable
• We use the relationship Cij – Ui + Vj = 0 for x11 to calculate V1.
• C11 – U1 + V1 = 0 => 6 – 0 + V1 = 0 • => V1 = -6
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
3
Dummy
1 2
New York Montreal
0
5000
0
5000
01
2
3 Worcester0 20000
Boston15000 0
Hartford10000
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
3
Dummy
1 2
New York Montreal
-6
0
5000
0
5000
01
2
3 Worcester0 20000
Boston15000 0
Hartford10000
VAM
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 29
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
3
Dummy
1 2
New York Montreal
-6
0
5000
0
5000
01
2
3 Worcester0 20000
Boston15000 0
Hartford10000
The Transportation Simplex Method – Identify Entering
Variable
• We use the relationship Cij – Ui + Vj = 0 for x21 to calculate U2.
• C21 – U2 + V1 = 0 => 4 – U2 + (-6) = 0 • => U2 = -2
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
3
Dummy
1 2
New York Montreal
-6
0
5000
0
5000
0
-2
1
2
3 Worcester0 20000
Boston15000 0
Hartford10000
VAM
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 30
The Transportation Simplex Method – Identify Entering
Variable
• We use the relationship Cij – Ui + Vj = 0 for x13 to calculate V3.
• C12 – U1 + V3 = 0 => 100 – 0 + V3 = 0 • => V3 = -100
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
3
Dummy
1 2
New York Montreal
-6
0
5000
0
5000
0
-2
1
2
3 Worcester0 20000
Boston15000 0
Hartford10000
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
-100
3
Dummy
1 2
New York Montreal
-6
0
5000
0
5000
0
-2
1
2
3 Worcester0 20000
Boston15000 0
Hartford10000
VAM
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 31
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
-100
3
Dummy
1 2
New York Montreal
-6
0
5000
0
5000
0
-2
1
2
3 Worcester0 20000
Boston15000 0
Hartford10000
The Transportation Simplex Method – Identify Entering
Variable
• We use the relationship Cij – Ui + Vj = 0 for x33 to calculate U3.
• C33 – U3 + V3 = 0 => 100 – U3 + (-100) = 0
• => U3 = 0
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100Worcester
0 20000
Boston15000 0
Hartford10000
0
-2
0
1
2
3
5000
0
5000
0
-100
3
Dummy
1 2
New York Montreal
-6
VAM
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 32
The Transportation Simplex Method – Identify Entering
Variable
• We use the relationship Cij – Ui + Vj = 0 for x32 to calculate V2.
• C32 – U3 + V2 = 0 => 10 – 0 + V2 = 0 • => 10 – 0 + V2 = 0 => -10 = V2
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100Worcester
0 20000
Boston15000 0
Hartford10000
0
-2
0
1
2
3
5000
0
5000
0
-100
3
Dummy
1 2
New York Montreal
-6
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100Worcester
0 20000
Boston15000 0
Hartford10000
0
-2
0
1
2
3
5000
0
5000
0
-100
3
Dummy
1 2
New York Montreal
-6 -10
VAM
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 33
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100Worcester
0 20000
Boston15000 0
Hartford10000
0
-2
0
1
2
3
5000
0
5000
0
-100
3
Dummy
1 2
New York Montreal
-6 -10
• We use the relationship Cij – Zij = Cij – Ui + Vj for x12 to calculate C12 – Z12.
• We use the relationship Cij – Zij = Cij – Ui + Vj for x22 to calculate C22 – Z22.
• C22 – Z22 = C22 – U2 + V2 • C22 – Z22 = 11 – (-2) + (-10) = 3
The Transportation Simplex Method – Identify Entering
Variable
• C12 – Z12 = C12 – U1 + V2 • C12 – Z12 = 12 – 0 + (-10) = 2• We use the relationship Cij – Zij = Cij – Ui + Vj
for x31 to calculate C31 – Z31.• We use the relationship Cij – Zij = Cij – Ui + Vj
for x23 to calculate C23 – Z23.• C23 – Z23 = C23 – U2 + V3 • C23 – Z23 = 100 – (-2) + (-100) = 2• C31 – Z31 = C31 – U3 + V1 • C31 – Z31 = 5 – 0 + (-6) = -1
Cij - Zij
x12 2
x22 3
x23 2
x31 -1
Cij - Zij
x12 2
x22 3
x23 2
x31 -1
Cij - Zij
x12 2
x22 3
x23 2
x31 -1
Cij - Zij
x12 2
x22 3
x23 2
x31 -1
x31 is the
Entering Variable!
VAM
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 34
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100Worcester
0 20000
Boston15000 0
Hartford10000
0
-2
0
1
2
3
5000
0
5000
0
-100
3
Dummy
1 2
New York Montreal
-6 -10
• Label the Entering Variable with a “+” and the other variables around the Cycle “-”, “+” and “-”.
• We look for a “Cycle” of at least 3 Basic Variables and the Entering Variable.
The Transportation Simplex Pivot
+-
+ -
Cij - Zij
x12 2
x22 3
x23 2
x31 -1
x31 is the
Entering Variable!
VAM
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 35
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100Worcester
0 20000
Boston15000 0
Hartford10000
0
-2
0
1
2
3
5000
0
5000
0
-100
3
Dummy
1 2
New York Montreal
-6 -10
The Transportation Simplex Pivot
• Then work out the max x31 can be increased by without decreasing x11 or x33 below 0. i.e. Min(x11, x33) = 5000
+ -
+-
VAM
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 36
The Transportation Simplex Pivot
• Increase x31 and x13 by 5,000.• Decrease x11 and x33 by 5,000.
VAM
Cij - Zij
x12 1
x21 2
x23 2
x33 1
Cij - Zij
x12 1
x21 2
x23 2
x33 1
Cij - Zij
x12 1
x21 2
x23 2
x33 1
Cij - Zij
x12 1
x21 2
x23 2
x33 1
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100Worcester
0 20000
Boston15000 0
Hartford10000
0
-2
0
1
2
3
5000
0
5000
0
-100
3
Dummy
1 2
New York Montreal
-6 -10
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
0
5000
3
-100
Dummy
-6 -11
New York Montreal
1 2
1 Boston15000 0
0
2 Hartford10000 0
-2
3 -1 Worcester5000 20000
10000
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
0
0
3
-100
Dummy
-6 -11
New York Montreal
1 2
1 Boston10000 0
0
C
2 Hartford10000 0
-2
3 -1 Worcester5000 20000
10000
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
0
0
3
-100
Dummy
-6 -11
New York Montreal
1 2
1 Boston10000 0
0
C
2 Hartford10000 0
-2
3 -1 Worcester5000 20000
10000
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
0
0
3
-100
Dummy
-6 -11
New York Montreal
1 2
1 Boston10000 0
0
C
2 Hartford10000 0
-2
3 -1 Worcester5000 20000
10000
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
0
0
3
-100
Dummy
-6 -11
New York Montreal
1 2
1 Boston10000 0
0
C
2 Hartford10000 0
-2
3 -1 Worcester5000 20000
10000
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
0
0
3
-100
Dummy
-6 -11
New York Montreal
1 2
1 Boston10000 0
0
C
2 Hartford10000 0
-2
3 -1 Worcester5000 20000
10000
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
0
0
3
-100
Dummy
-6 -11
New York Montreal
1 2
1 Boston10000 0
0
C
2 Hartford10000 0
-2
3 -1 Worcester5000 20000
10000
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 37
The Transportation Simplex
• We have seen this before! • It is the optimal solution!!
• We ship 10,000 from Boston to New York, with 10,000 slack capacity in Boston. We ship 10,000 from Hartford to New York, 5,000 from Worcester to New York and 20,000 from Worcester to Montreal.
Cij - Zij
x12 1
x21 2
x23 2
x33 1
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
0
0
3
-100
Dummy
-6 -11
New York Montreal
1 2
1 Boston10000 0
0
C
2 Hartford10000 0
-2
3 -1 Worcester5000 20000
10000
VAM
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 38
• If we look at the result of applying the North West Corner method to derive a Basic Feasible Solution.
• We will see in the following slides, that we need to undertake a more complex pivot involving a cycle which has more than four elements…
A More Complex Pivot
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 39
• We look for a “Cycle” of at least 3 Basic Variables and the Entering Variable.
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100Worcester
0 15000
Boston20000 0
Hartford5000
0
-2
0
1
2
3
0
0
10000
5000
-100
3
Dummy
1 2
New York Montreal
-6 -13
• A cycle which includes the four corners looks good, but if we use this we get 6 Basic Variables!
A More Complex Pivot
Cij - Zij
x12 -1
x13 -3
x23 -1
x31 2
x13 is the
Entering Variable!
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
1 2 3
-6 -13 -100
New York Montreal Dummy
1 0 Boston10000 0 10000
2 -2 Hartford5000 5000 0
3 0 Worcester10000 15000 0
NWC
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 40
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100Worcester
0 15000
Boston20000 0
Hartford5000
0
-2
0
1
2
3
0
0
10000
5000
-100
3
Dummy
1 2
New York Montreal
-6 -13
• So we need to find a “Cycle” of at least 5 Basic Variables and the Entering Variable.
A More Complex Pivot
+
+
+
-
-
-5000
• We then need to find the minimum leaving variable…
• So, add 5000 to x13, x32, x21 and deduct 5000 from x33, x22 and x11 to complete the pivot.
Remember x13 is
the Entering Variable!
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100
5000
Dummy
5000
0
3
-100
1 2
-6 -10
New York Montreal
1 0 Boston15000 0
2 -2 Hartford10000
3 0 Worcester0
0
20000
NWC
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 41
j =
Vj
i Ui
6 12 100
4 11 100
5 10 100Worcester
0 15000
Boston20000 0
Hartford5000
0
-2
0
1
2
3
0
0
10000
5000
-100
3
Dummy
1 2
New York Montreal
-6 -13
• This time we can find a simpler pivot involving 3 Basic Variables and the Entering Variable.
A More Complex Pivot
Cij - Zij
x12 2
x22 3
x23 2
x31 -1
+
- +
-5000
j =
Vj
i Ui
6 12 100
4 11 100
5 10 1003 -1 Worcester
5000 20000 0
2 -2 Hartford10000 0 0
10000 0 10000
Dummy
1 0 Boston
1 2 3
-6 -11 -100
New York Montreal
NWC
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 42
The Transportation Simplex
• We have seen this before! • It is the optimal solution!!
• We ship 10,000 from Boston to New York, with 10,000 slack capacity in Boston. We ship 10,000 from Hartford to New York, 5,000 from Worcester to New York and 20,000 from Worcester to Montreal.
j =
Vj
i Ui
6 12 100
4 11 100
5 10 1003 -1 Worcester
5000 20000 0
2 -2 Hartford10000 0 0
10000 0 10000
Dummy
1 0 Boston
1 2 3
-6 -11 -100
New York MontrealCij - Zij
x12 1
x21 2
x23 2
x33 1
NWC
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to Management Science
Lecture 18 Slide 43
Reading and Homework.
• Read Network Model Algorithms Supplement 5 Section III
• Read the Assignment Problem Handout.