MEIO/UPC-UB : NETWORK FLOWS
Network Flows
UPCOPENCOURSEWARE number 34414
Topic 1: Introduction
F.-Javier Heredia
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• Definitions:
– Minimum shortest path problems.
– Maximum flow problems.
– Minimum cost flow problems.
• Network flow problems (NF) as a class of linear
programming problems (LP):
–
–
–
–
–
–
Standard form of the minimum flow cost problem.
Nodes-arcs incidence matrix.
Formulation of the minimum shortest path problem.
Formulation of the maximum flow problem.
Formulation of the transportation and assignment problem.
Transformation to the standard form.
• Applications.
F.-Javier Heredia http://gnom.upc.edu/heredia
Introduction-2
introduction
MEIO/UPC-UB : NETWORK FLOWS
Introduction
• To Identify the shortest path between a source node
(“s”) and the sink node (“t”).
20
2
4
10
30
s
15
1
35
t
40
20
25
3
• Applications:
6
35
5
– Minimum cost flow problem.
– Minimum time flow problem.
– Equipment replacement problems.
F.-Javier Heredia http://gnom.upc.edu/heredia
Introduction-3
introduction
MEIO/UPC-UB : NETWORK FLOWS
Shortest Path Problems
• To seek how to send the maximum possible amount of
flow from a source node (“s”) to sink node (“t”) in a
capacitated network.
3
3
2
6
2 2
5
s
4
5
6
t
1
6
4
4
2
4
2
3
2
5
4
5
• Applications: determining the maximum steady –state
flow of:
–
–
–
–
Petroleum products in a pipeline network.
Cars in a road network.
Messages in a telecommunication network.
Electricity in an electrical network.
F.-Javier Heredia http://gnom.upc.edu/heredia
Introduction-4
introduction
MEIO/UPC-UB : NETWORK FLOWS
Maximum Flow Problems
• We wish to determie a least shipment cost of a single commodity
through a network in order to satisfy comsuption at demand
nodes with the production of the supply nodes.
10
2
-15
2
4
3
5
5 1
4
10
3
• Applications:
5
6
3
0
1
5
0
2
4
7
10
6
6
-10
– Logistics(warehouses to retailers).
– Automobile routing in an urban traffic network.
– Routing of calls through a telephone system.
F.-Javier Heredia http://gnom.upc.edu/heredia
Introduction-5
introduction
MEIO/UPC-UB : NETWORK FLOWS
Minimum Flow Cost Problems
• Let G=(N,A) be the directed graph defined by the set N of n nodes
(|N| = n) and the set A of m directed arcs (|A| = m) .
10
2
2
3
5
5
4
1
b4 = -15
4
1
5
6
3
(7,4)
c74 = 2
7
4
b7 = +10
6
3
0
5
6
-10
n=7 ; m=11 ; N={1,2,3,4,5,6,7} ; A={ (1,2), (1,3), (2,3),...}
• Demand/supply vector:
bj , j=1,2,...,n: b=[5,10,0,-15,0,5,10]’
• Cost vector: cij , (i,j)∈A : c=[5,3,4,...]’
• Flow: xij , (i,j)∈A : amout of commodity to be sent between node i
and node j through arc (i,j).
F.-Javier Heredia http://gnom.upc.edu/heredia
Introduction-6
introduction
MEIO/UPC-UB : NETWORK FLOWS
Standard form of the minimum
cost flow problem (MCNFP) (I)
• Mathematic formulation:
 min

s.a. :






z=
∑c x
(i,j)∈ A
∑ xij −
{ j:( i , j )∈A}
ij ij
(1.1a)
∑ x ji = bi ∀i ∈ N (1.1b)
{ j:( j ,i )∈A}
xij ≥ 0 ∀(i, j ) ∈ A
Balance
equations
(1.1c)
n
– We assume the network is balanced:
∑b = 0
i =1
• Matrix notation:
i
c′x
(1.2a)
min

s.a. : T x = b (1.2b)

x ≥ 0 (1.2c)

F.-Javier Heredia http://gnom.upc.edu/heredia
Introduction-7
introduction
MEIO/UPC-UB : NETWORK FLOWS
Standard form of the minimum
cost
cost flow
flow problem
problem(II)
(II)
• The associated node-arc
• Consider the
incidence matrix T is:
following network:
4
(35,50)
3
i
(25,20)
(cij ,uij)
(1, 2 ) (1,3) ( 2, 4 ) ( 3, 2 ) ( 4,3) ( 4,5) ( 5,3) ( 5, 4 )
1  1 1 0 0 0 0 0 0
2 − 1 0
1 − 1 0 0 0 0
T= 
3 0 −1 0
1 − 1 0 − 1 0


4  0 0 −1 0
1 1 0 − 1
5  0 0 0 0 0 − 1 1 1
• Properties:
5
j
–
–
–
–
2m non-zero elements among nm.
Only 2 elements ≠0 per column.
Every non-zero elements is +1 or -1.
Rank(T)= n-1.
F.-Javier Heredia http://gnom.upc.edu/heredia
Introduction-8
introduction
1
(15,40)
(45,60)
2
45,10)
MEIO/UPC-UB : NETWORK FLOWS
Node-arc incidence matrix
20
2
4
10
cij
i
j
30
+1
15
1
35
-1
40
6
20
25
3
35
5
• b1= +1 ; b6= -1 ; bj= 0 ∀j ≠1,6
F.-Javier Heredia http://gnom.upc.edu/heredia
Introduction-9
introduction
MEIO/UPC-UB : NETWORK FLOWS
Shortest path problem formulated as a MCNFP
2
(0,3)
4
i
s
1
(cij ,uij)
j
t
6
3
(0,5)
5
(-1,+∞)
• Artificial arc xts with cts=-1 and uts= +∞.
• cij=0 ∀(i,j)≠(t,s)
F.-Javier Heredia http://gnom.upc.edu/heredia
Introduction-10
introduction
MEIO/UPC-UB : NETWORK FLOWS
Maximum flow problem formulated as a MCNFP
1
5
2
6
3
4
7
• Properties:
– N = N1∪ N2:
N1: production nodes; N2: demand nodes.
– ∀(i,j)∈A : i ∈ N1 ; j ∈ N2
F.-Javier Heredia http://gnom.upc.edu/heredia
Introduction-11
introduction
MEIO/UPC-UB : NETWORK FLOWS
Transportation problem formulated as a MCNFP
+1
1
4
+1
-1
-1
2
5
+1
-1
3
6
• Properties:
– N = N1∪ N2 ; |N1 |= |N2|
– A⊆ N1× N2.
– bj=+1 ∀j ∈N1 ; bj=-1 ∀j ∈N2
F.-Javier Heredia http://gnom.upc.edu/heredia
Introduction-12
introduction
MEIO/UPC-UB : NETWORK FLOWS
Assigment problem formulation
• Undirected arcs:
xij
j
i
xji
– Contribution to the objective function: cij xij +cij xji
– cij ≥ 0 ⇒ at the optimal solution xij > 0 or xji > 0
– Transformation: each directed arc {i,j} is replaced by
two directed arcs (i,j) and (j,i) with cost cij :
xij
j
i
xji
F.-Javier Heredia http://gnom.upc.edu/heredia
Introduction-13
introduction
MEIO/UPC-UB : NETWORK FLOWS
Transformation to the standard form (I)
• Arcs with non-zero lower bounds:
– xij ≥ lij
– xij is replaced by x’ij+lij in the problem’s formulation:
 Bounds: xij ≥ lij
 Balance
⇒ x’ij+lij ≥ lij ; x’ij ≥ 0
equations: bi → bi - lij
 Objective
; bj → bj + lij
function: cij xij → cij (x’ij + lij ) = cij x’ij + cij lij
cte.
F.-Javier Heredia http://gnom.upc.edu/heredia
Introduction-14
introduction
MEIO/UPC-UB : NETWORK FLOWS
Transformation to the standard form (II)
• Arcs with negative costs:
– Let be xij with cij<0.
– Let uij a trivial upper bound of the arc (i,j).
– xij is replaced by x’ij = uij – xij.
 0 ≤ x’ij ≤ uij
cst >0
function:
z = ...+ cij xij +...→ z’= ...+ cijuij – cij x’ij +...
 Objective
 Balance
bi
equations:
xij
i
cij ,uij
bj
j
bi - uij x’ bj + uij
ij
i
j
-cij ,uij
F.-Javier Heredia http://gnom.upc.edu/heredia
Introduction-15
introduction
MEIO/UPC-UB : NETWORK FLOWS
Transformation to the standard form (III)
• Arcs with capacity:
– Let 0 ≤ xij ≤ uij.
– Network transformation:
bi
i
xij
cij ,uij
 The
bj
j
bi
xik
i c ,∞
ik
- uij
k
xjk
0 ,∞
bj + uij
j
network is equivalent to the original:
xij ≡ xik ; xik + xjk = uij ⇒ xik = xij ≤ uij
 The objective function id not modified: cik ≡ cij
F.-Javier Heredia http://gnom.upc.edu/heredia
Introduction-16
introduction
MEIO/UPC-UB : NETWORK FLOWS
Transformation to the standard form (IV)
• Unbalanced network:
– Excess of production
n
(
∑b
j =1
j
– Excess of demand
n
>0 ):
(
∑b
j =1
+10
1
3
-8
j
<0 ):
+10
1
3
-12
+4
-3
5
5
+8
2
4
-7
+8
2
4
-10
A dummy demand node n+1 is
A dummy supply node n+1 is
added linking all production nodes
added, linking all demand nodes
through uncapacitated - null
through uncapacitated – null
cost arcs.
cost arcs.
F.-Javier Heredia http://gnom.upc.edu/heredia
Introduction-17
introduction
MEIO/UPC-UB : NETWORK FLOWS
Transformation to the standard form (V)
• Example 1:
bj
bi
(cij ,uij ,lij )
j
i
-15
20
1
-10
(-2,10,0)
4
(2, ∞,0)
2
(-1,20,0)
(10, ∞,5)
(Solution)
3
5
10
0
F.-Javier Heredia http://gnom.upc.edu/heredia
Introduction-18
introduction
MEIO/UPC-UB : NETWORK FLOWS
Transformation to the standard form (VI)
• Solve the applications examples from book
chapter 1.3 of AMO (1 week):
– Problem description.
– Network formulation and objective function
associated to the problem.
– Problem classification.
• Problems assigment:
F.-Javier Heredia http://gnom.upc.edu/heredia
Introduction-19
introduction
MEIO/UPC-UB : NETWORK FLOWS
Applications
1. Negative cost elimination
b
bi
(cij ,uij ,lij ) j
j
i
-15-10-20= -45
20
4
z=z’+20+20=z’+40
(10, ∞,5)
3
10+20 = +30
5
0
F.-Javier Heredia http://gnom.upc.edu/heredia
Introduction-20
introduction
1
-10+10 = 0
(2,10,0)
(2, ∞,0)
2
(1,20,0)
MEIO/UPC-UB : NETWORK FLOWS
Transformation to the standard form
Example (1/4)
2. Lower bounds elimination ≠0
b
bi
(cij ,uij ,lij ) j
j
i
0
-45+5 = -40
(2,10,0)
1
z=z’+40+25+50=z’+115
(10, ∞,0)
3
30-5 = -25
5
0+5 = 5
F.-Javier Heredia http://gnom.upc.edu/heredia
Introduction-21
introduction
20-5 = 15
4
(2, ∞,0)
2
(1,20,0)
MEIO/UPC-UB : NETWORK FLOWS
Transformation to the standard form
Example (2/4)
3. Capacity elimination
bj
bi
(cij ,uij ,lij )
j
i
-40+20+10 = 30
(2, ∞,0) 0
(0, ∞,0)
7
2
4
z=z’+115
-10
6
1
-20
(10, ∞,0)
3
5
-25
5
F.-Javier Heredia http://gnom.upc.edu/heredia
Introduction-22
introduction
15
(2, ∞,0)
MEIO/UPC-UB : NETWORK FLOWS
Transformation to the standard form
Example (3/4)
4. Unbalance network
bj
bi
(cij ,uij ,lij )
j
i
z=z’+115
30
(2, ∞,0) 0
(0, ∞,0)
7
2
4
-10
6
1
-20
(10, ∞,0)
3
5
5
-25
F.-Javier Heredia http://gnom.upc.edu/heredia
Introduction-23
introduction
-25
15
7
(2, ∞,0)
MEIO/UPC-UB : NETWORK FLOWS
Transformation to the standard form
Example (4/4)