Linear Programming: Chapter 13 Network Flows: Theoryanitescu/TEACHING/ATKINSON/m1100new.dir/… ·...
Transcript of Linear Programming: Chapter 13 Network Flows: Theoryanitescu/TEACHING/ATKINSON/m1100new.dir/… ·...
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Linear Programming: Chapter 13Network Flows: Theory
Robert J. Vanderbei
May 21, 2000
Operations Research and Financial Engineering
Princeton University
Princeton, NJ 08544
http://www.princeton.edu/∼rvdb
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1 Networks
Basic elements:
• N Nodes (let m denote number of them).
• A Directed Arcs
– subset of all possible arcs: {(i, j) : i, j ∈ N , i 6= j}.
– arcs are directed: (i, j) 6= (j, i).
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1 Networks
Basic elements:
• N Nodes (let m denote number of them).
• A Directed Arcs
– subset of all possible arcs: {(i, j) : i, j ∈ N , i 6= j}.
– arcs are directed: (i, j) 6= (j, i).
![Page 4: Linear Programming: Chapter 13 Network Flows: Theoryanitescu/TEACHING/ATKINSON/m1100new.dir/… · Linear Programming: Chapter 13 Network Flows: Theory Robert J. Vanderbei May 21,](https://reader035.fdocuments.us/reader035/viewer/2022070901/5f4482a0920ba45dec021cb2/html5/thumbnails/4.jpg)
1 Networks
Basic elements:
• N Nodes (let m denote number of them).
• A Directed Arcs
– subset of all possible arcs: {(i, j) : i, j ∈ N , i 6= j}.
– arcs are directed: (i, j) 6= (j, i).
![Page 5: Linear Programming: Chapter 13 Network Flows: Theoryanitescu/TEACHING/ATKINSON/m1100new.dir/… · Linear Programming: Chapter 13 Network Flows: Theory Robert J. Vanderbei May 21,](https://reader035.fdocuments.us/reader035/viewer/2022070901/5f4482a0920ba45dec021cb2/html5/thumbnails/5.jpg)
1 Networks
Basic elements:
• N Nodes (let m denote number of them).
• A Directed Arcs
– subset of all possible arcs: {(i, j) : i, j ∈ N , i 6= j}.
– arcs are directed: (i, j) 6= (j, i).
![Page 6: Linear Programming: Chapter 13 Network Flows: Theoryanitescu/TEACHING/ATKINSON/m1100new.dir/… · Linear Programming: Chapter 13 Network Flows: Theory Robert J. Vanderbei May 21,](https://reader035.fdocuments.us/reader035/viewer/2022070901/5f4482a0920ba45dec021cb2/html5/thumbnails/6.jpg)
1 Networks
Basic elements:
• N Nodes (let m denote number of them).
• A Directed Arcs
– subset of all possible arcs: {(i, j) : i, j ∈ N , i 6= j}.
– arcs are directed: (i, j) 6= (j, i).
![Page 7: Linear Programming: Chapter 13 Network Flows: Theoryanitescu/TEACHING/ATKINSON/m1100new.dir/… · Linear Programming: Chapter 13 Network Flows: Theory Robert J. Vanderbei May 21,](https://reader035.fdocuments.us/reader035/viewer/2022070901/5f4482a0920ba45dec021cb2/html5/thumbnails/7.jpg)
2 Network Flow Data
• bi, i ∈ N , supply at node i
• cij, (i, j) ∈ A, cost of shipping 1 unit along arc (i, j).
Note: demands are recorded as negative supplies.
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2 Network Flow Data
• bi, i ∈ N , supply at node i
• cij, (i, j) ∈ A, cost of shipping 1 unit along arc (i, j).
Note: demands are recorded as negative supplies.
![Page 9: Linear Programming: Chapter 13 Network Flows: Theoryanitescu/TEACHING/ATKINSON/m1100new.dir/… · Linear Programming: Chapter 13 Network Flows: Theory Robert J. Vanderbei May 21,](https://reader035.fdocuments.us/reader035/viewer/2022070901/5f4482a0920ba45dec021cb2/html5/thumbnails/9.jpg)
2 Network Flow Data
• bi, i ∈ N , supply at node i
• cij, (i, j) ∈ A, cost of shipping 1 unit along arc (i, j).
Note: demands are recorded as negative supplies.
![Page 10: Linear Programming: Chapter 13 Network Flows: Theoryanitescu/TEACHING/ATKINSON/m1100new.dir/… · Linear Programming: Chapter 13 Network Flows: Theory Robert J. Vanderbei May 21,](https://reader035.fdocuments.us/reader035/viewer/2022070901/5f4482a0920ba45dec021cb2/html5/thumbnails/10.jpg)
2 Network Flow Data
• bi, i ∈ N , supply at node i
• cij, (i, j) ∈ A, cost of shipping 1 unit along arc (i, j).
Note: demands are recorded as negative supplies.
![Page 11: Linear Programming: Chapter 13 Network Flows: Theoryanitescu/TEACHING/ATKINSON/m1100new.dir/… · Linear Programming: Chapter 13 Network Flows: Theory Robert J. Vanderbei May 21,](https://reader035.fdocuments.us/reader035/viewer/2022070901/5f4482a0920ba45dec021cb2/html5/thumbnails/11.jpg)
3 Network Flow Problem
Decision Variables:
• xij, (i, j) ∈ A, quantity to ship along arc (i, j).
Objective:
minimize∑
(i,j)∈A
cijxij
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3 Network Flow Problem
Decision Variables:
• xij, (i, j) ∈ A, quantity to ship along arc (i, j).
Objective:
minimize∑
(i,j)∈A
cijxij
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4 Network Flow Problem–Cont.
Constraints:
• Mass conservation (aka flow balance):
inflow(k) − outflow(k) = demand(k) = −supply(k), k ∈ N
m∑i :
(i, k) ∈ A
xik −∑j :
(k, j) ∈ A
xkj = −bk, k ∈ N
• Nonnegativity:
xij ≥ 0, (i, j) ∈ A
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4 Network Flow Problem–Cont.
Constraints:
• Mass conservation (aka flow balance):
inflow(k) − outflow(k) = demand(k) = −supply(k), k ∈ N
m∑i :
(i, k) ∈ A
xik −∑j :
(k, j) ∈ A
xkj = −bk, k ∈ N
• Nonnegativity:
xij ≥ 0, (i, j) ∈ A
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4 Network Flow Problem–Cont.
Constraints:
• Mass conservation (aka flow balance):
inflow(k) − outflow(k) = demand(k) = −supply(k), k ∈ N
m∑i :
(i, k) ∈ A
xik −∑j :
(k, j) ∈ A
xkj = −bk, k ∈ N
• Nonnegativity:
xij ≥ 0, (i, j) ∈ A
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5 Matrix Notation
minimize cT x
subject to Ax = −b
x ≥ 0
where
cT =[
2 4 9 11 4 3 8 7 0 15 16 18]
A =
−1 −1 1 1 1
1 1 1
−1 −1
1 1 1 1
−1 −1 −1 1
−1 −1 1
−1 −1 −1
, b =
0
−19
16
−33
0
0
36
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6 Notes
• A is called node-arc incidence matrix.
• A is large and sparse.
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6 Notes
• A is called node-arc incidence matrix.
• A is large and sparse.
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6 Notes
• A is called node-arc incidence matrix.
• A is large and sparse.
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7 Dual Problem
maximize −bT y
subject to AT y + z = c
z ≥ 0
In network notation:
maximize −∑
i∈N biyi
subject to yj − yi + zij = cij (i, j) ∈ A
zij ≥ 0 (i, j) ∈ A
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8 Complementarity Relations
• The primal variables must be nonnegative.
• Therefore the associated dual constraints are inequalities.
• The dual slack variables are complementary to the primal variables:
xijzij = 0, (i, j) ∈ A
• The primal constraints are equalities.
• Therefore they have no slack variables.
• The corresponding dual variables, the yi’s, are free variables.
• No complementarity conditions apply to them.
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8 Complementarity Relations
• The primal variables must be nonnegative.
• Therefore the associated dual constraints are inequalities.
• The dual slack variables are complementary to the primal variables:
xijzij = 0, (i, j) ∈ A
• The primal constraints are equalities.
• Therefore they have no slack variables.
• The corresponding dual variables, the yi’s, are free variables.
• No complementarity conditions apply to them.
![Page 23: Linear Programming: Chapter 13 Network Flows: Theoryanitescu/TEACHING/ATKINSON/m1100new.dir/… · Linear Programming: Chapter 13 Network Flows: Theory Robert J. Vanderbei May 21,](https://reader035.fdocuments.us/reader035/viewer/2022070901/5f4482a0920ba45dec021cb2/html5/thumbnails/23.jpg)
8 Complementarity Relations
• The primal variables must be nonnegative.
• Therefore the associated dual constraints are inequalities.
• The dual slack variables are complementary to the primal variables:
xijzij = 0, (i, j) ∈ A
• The primal constraints are equalities.
• Therefore they have no slack variables.
• The corresponding dual variables, the yi’s, are free variables.
• No complementarity conditions apply to them.
![Page 24: Linear Programming: Chapter 13 Network Flows: Theoryanitescu/TEACHING/ATKINSON/m1100new.dir/… · Linear Programming: Chapter 13 Network Flows: Theory Robert J. Vanderbei May 21,](https://reader035.fdocuments.us/reader035/viewer/2022070901/5f4482a0920ba45dec021cb2/html5/thumbnails/24.jpg)
8 Complementarity Relations
• The primal variables must be nonnegative.
• Therefore the associated dual constraints are inequalities.
• The dual slack variables are complementary to the primal variables:
xijzij = 0, (i, j) ∈ A
• The primal constraints are equalities.
• Therefore they have no slack variables.
• The corresponding dual variables, the yi’s, are free variables.
• No complementarity conditions apply to them.
![Page 25: Linear Programming: Chapter 13 Network Flows: Theoryanitescu/TEACHING/ATKINSON/m1100new.dir/… · Linear Programming: Chapter 13 Network Flows: Theory Robert J. Vanderbei May 21,](https://reader035.fdocuments.us/reader035/viewer/2022070901/5f4482a0920ba45dec021cb2/html5/thumbnails/25.jpg)
8 Complementarity Relations
• The primal variables must be nonnegative.
• Therefore the associated dual constraints are inequalities.
• The dual slack variables are complementary to the primal variables:
xijzij = 0, (i, j) ∈ A
• The primal constraints are equalities.
• Therefore they have no slack variables.
• The corresponding dual variables, the yi’s, are free variables.
• No complementarity conditions apply to them.
![Page 26: Linear Programming: Chapter 13 Network Flows: Theoryanitescu/TEACHING/ATKINSON/m1100new.dir/… · Linear Programming: Chapter 13 Network Flows: Theory Robert J. Vanderbei May 21,](https://reader035.fdocuments.us/reader035/viewer/2022070901/5f4482a0920ba45dec021cb2/html5/thumbnails/26.jpg)
8 Complementarity Relations
• The primal variables must be nonnegative.
• Therefore the associated dual constraints are inequalities.
• The dual slack variables are complementary to the primal variables:
xijzij = 0, (i, j) ∈ A
• The primal constraints are equalities.
• Therefore they have no slack variables.
• The corresponding dual variables, the yi’s, are free variables.
• No complementarity conditions apply to them.
![Page 27: Linear Programming: Chapter 13 Network Flows: Theoryanitescu/TEACHING/ATKINSON/m1100new.dir/… · Linear Programming: Chapter 13 Network Flows: Theory Robert J. Vanderbei May 21,](https://reader035.fdocuments.us/reader035/viewer/2022070901/5f4482a0920ba45dec021cb2/html5/thumbnails/27.jpg)
8 Complementarity Relations
• The primal variables must be nonnegative.
• Therefore the associated dual constraints are inequalities.
• The dual slack variables are complementary to the primal variables:
xijzij = 0, (i, j) ∈ A
• The primal constraints are equalities.
• Therefore they have no slack variables.
• The corresponding dual variables, the yi’s, are free variables.
• No complementarity conditions apply to them.
![Page 28: Linear Programming: Chapter 13 Network Flows: Theoryanitescu/TEACHING/ATKINSON/m1100new.dir/… · Linear Programming: Chapter 13 Network Flows: Theory Robert J. Vanderbei May 21,](https://reader035.fdocuments.us/reader035/viewer/2022070901/5f4482a0920ba45dec021cb2/html5/thumbnails/28.jpg)
8 Complementarity Relations
• The primal variables must be nonnegative.
• Therefore the associated dual constraints are inequalities.
• The dual slack variables are complementary to the primal variables:
xijzij = 0, (i, j) ∈ A
• The primal constraints are equalities.
• Therefore they have no slack variables.
• The corresponding dual variables, the yi’s, are free variables.
• No complementarity conditions apply to them.
![Page 29: Linear Programming: Chapter 13 Network Flows: Theoryanitescu/TEACHING/ATKINSON/m1100new.dir/… · Linear Programming: Chapter 13 Network Flows: Theory Robert J. Vanderbei May 21,](https://reader035.fdocuments.us/reader035/viewer/2022070901/5f4482a0920ba45dec021cb2/html5/thumbnails/29.jpg)
9 Definition: Subnetwork
Network
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9 Definition: Subnetwork
Network Subnetwork
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10 Connected vs. Disconnected
Connected
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10 Connected vs. Disconnected
Connected Disconnected
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11 Cyclic vs. Acyclic
Cyclic
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11 Cyclic vs. Acyclic
Cyclic Acyclic
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12 Trees
Tree = Connected + Acyclic
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12 Trees
Tree = Connected + Acyclic Not Trees
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13 Spanning Trees
Spanning Tree–A tree touching every node
Tree Solution
xij = 0 for (i, j) 6∈ Tree Arcs
Note: Tree solutions are easy to compute—start at the leaves and work inward...
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13 Spanning Trees
Spanning Tree–A tree touching every node
Tree Solution
xij = 0 for (i, j) 6∈ Tree Arcs
Note: Tree solutions are easy to compute—start at the leaves and work inward...
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13 Spanning Trees
Spanning Tree–A tree touching every node
Tree Solution
xij = 0 for (i, j) 6∈ Tree Arcs
Note: Tree solutions are easy to compute—start at the leaves and work inward...
![Page 40: Linear Programming: Chapter 13 Network Flows: Theoryanitescu/TEACHING/ATKINSON/m1100new.dir/… · Linear Programming: Chapter 13 Network Flows: Theory Robert J. Vanderbei May 21,](https://reader035.fdocuments.us/reader035/viewer/2022070901/5f4482a0920ba45dec021cb2/html5/thumbnails/40.jpg)
14 Online Pivot Tool–Notations
Data:
• Costs on arcs shown above arcs.
• Supplies at nodes shown above nodes.
Variables:
• Primal flows shown on tree arcs.
• Dual slacks shown on nontree arcs.
• Dual variables shown below nodes.
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14 Online Pivot Tool–Notations
Data:
• Costs on arcs shown above arcs.
• Supplies at nodes shown above nodes.
Variables:
• Primal flows shown on tree arcs.
• Dual slacks shown on nontree arcs.
• Dual variables shown below nodes.
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14 Online Pivot Tool–Notations
Data:
• Costs on arcs shown above arcs.
• Supplies at nodes shown above nodes.
Variables:
• Primal flows shown on tree arcs.
• Dual slacks shown on nontree arcs.
• Dual variables shown below nodes.
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14 Online Pivot Tool–Notations
Data:
• Costs on arcs shown above arcs.
• Supplies at nodes shown above nodes.
Variables:
• Primal flows shown on tree arcs.
• Dual slacks shown on nontree arcs.
• Dual variables shown below nodes.
![Page 44: Linear Programming: Chapter 13 Network Flows: Theoryanitescu/TEACHING/ATKINSON/m1100new.dir/… · Linear Programming: Chapter 13 Network Flows: Theory Robert J. Vanderbei May 21,](https://reader035.fdocuments.us/reader035/viewer/2022070901/5f4482a0920ba45dec021cb2/html5/thumbnails/44.jpg)
14 Online Pivot Tool–Notations
Data:
• Costs on arcs shown above arcs.
• Supplies at nodes shown above nodes.
Variables:
• Primal flows shown on tree arcs.
• Dual slacks shown on nontree arcs.
• Dual variables shown below nodes.
![Page 45: Linear Programming: Chapter 13 Network Flows: Theoryanitescu/TEACHING/ATKINSON/m1100new.dir/… · Linear Programming: Chapter 13 Network Flows: Theory Robert J. Vanderbei May 21,](https://reader035.fdocuments.us/reader035/viewer/2022070901/5f4482a0920ba45dec021cb2/html5/thumbnails/45.jpg)
14 Online Pivot Tool–Notations
Data:
• Costs on arcs shown above arcs.
• Supplies at nodes shown above nodes.
Variables:
• Primal flows shown on tree arcs.
• Dual slacks shown on nontree arcs.
• Dual variables shown below nodes.
![Page 46: Linear Programming: Chapter 13 Network Flows: Theoryanitescu/TEACHING/ATKINSON/m1100new.dir/… · Linear Programming: Chapter 13 Network Flows: Theory Robert J. Vanderbei May 21,](https://reader035.fdocuments.us/reader035/viewer/2022070901/5f4482a0920ba45dec021cb2/html5/thumbnails/46.jpg)
15 Tree Solutions–An Example
data
variables
• Fix a root node, say a.
• Primal flows on tree arcs calculated re-
cursively from leaves inward.
• Dual variables at nodes calculated recur-
sively from root node outward along tree
arcs using:
yj − yi = cij
• Dual slacks on nontree arcs calculated
using:
zij = yi − yj + cij.
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15 Tree Solutions–An Example
data
variables
• Fix a root node, say a.
• Primal flows on tree arcs calculated re-
cursively from leaves inward.
• Dual variables at nodes calculated recur-
sively from root node outward along tree
arcs using:
yj − yi = cij
• Dual slacks on nontree arcs calculated
using:
zij = yi − yj + cij.
![Page 48: Linear Programming: Chapter 13 Network Flows: Theoryanitescu/TEACHING/ATKINSON/m1100new.dir/… · Linear Programming: Chapter 13 Network Flows: Theory Robert J. Vanderbei May 21,](https://reader035.fdocuments.us/reader035/viewer/2022070901/5f4482a0920ba45dec021cb2/html5/thumbnails/48.jpg)
15 Tree Solutions–An Example
data
variables
• Fix a root node, say a.
• Primal flows on tree arcs calculated re-
cursively from leaves inward.
• Dual variables at nodes calculated recur-
sively from root node outward along tree
arcs using:
yj − yi = cij
• Dual slacks on nontree arcs calculated
using:
zij = yi − yj + cij.
![Page 49: Linear Programming: Chapter 13 Network Flows: Theoryanitescu/TEACHING/ATKINSON/m1100new.dir/… · Linear Programming: Chapter 13 Network Flows: Theory Robert J. Vanderbei May 21,](https://reader035.fdocuments.us/reader035/viewer/2022070901/5f4482a0920ba45dec021cb2/html5/thumbnails/49.jpg)
15 Tree Solutions–An Example
data
variables
• Fix a root node, say a.
• Primal flows on tree arcs calculated re-
cursively from leaves inward.
• Dual variables at nodes calculated recur-
sively from root node outward along tree
arcs using:
yj − yi = cij
• Dual slacks on nontree arcs calculated
using:
zij = yi − yj + cij.
![Page 50: Linear Programming: Chapter 13 Network Flows: Theoryanitescu/TEACHING/ATKINSON/m1100new.dir/… · Linear Programming: Chapter 13 Network Flows: Theory Robert J. Vanderbei May 21,](https://reader035.fdocuments.us/reader035/viewer/2022070901/5f4482a0920ba45dec021cb2/html5/thumbnails/50.jpg)
15 Tree Solutions–An Example
data
variables
• Fix a root node, say a.
• Primal flows on tree arcs calculated re-
cursively from leaves inward.
• Dual variables at nodes calculated recur-
sively from root node outward along tree
arcs using:
yj − yi = cij
• Dual slacks on nontree arcs calculated
using:
zij = yi − yj + cij.