1. Lecture 3 Transshipment Problems Minimum Cost Flow Problems1

2. Agenda
transshipment problems
minimum cost flow problems 2

3. Transshipment Problems3

4. Transshipment Problems
intermediate nodes C and D with flows passing through, neither created nor destroyed
minimum cost flows to send the goods through the nodes 9 4 11 6 1 2 5 8 3 C D F E G 60 40 35 45 20 A B 7 4

5. LP Formulation of Transshipment Problems
what are the decisions?
let xij be the amount of flow from node i to node j
objective:
min 7xAC + 4xAD + 9xBC + 11xBD xCE + 5xCF + 2xCG
+ xDE + 8xDF + 6xDG 9 4 11 6 1 2 5 8 3 C D F E G 60 40 35 45 20 A B 7 5

6. LP Formulation of Transshipment Problems
what are the rationale to set constraints?
non-negativity: xij  0  i, j
phenomena to model
related to the distribution of goods
equivalent to a valid flow pattern 9 4 11 6 1 2 5 8 3 C D F E G 60 40 35 45 20 A B 7
每一個限制式都是用符號和數學關係表達一個物理現象。 6

7. LP Formulation of Transshipment Problems
min objective of a flow pattern,
s.t. conditions to be a flow pattern. 9 4 11 6 1 2 5 8 3 C D F E G 60 40 35 45 20 A B 7
a valid flow pattern
 conservation of flows at all nodes
node A: xAC + xAD = 60
node B: xBC + xBD = 9
node C: xAC + xBC = xCE + xCF + xCG
node D: xAD + xBD = xDE + xDF + xDG
node E: xCE + xDE = 20
node F: xCF + xDF = 45
node G: xCG + xDG = 35 7

8. LP Formulation of Transshipment Problems
min 7xAC + 4xAD + 9xBC + 11xBD
+ 3xCE + 5xCF + 2xCG
+ xDE + 8xDF + 6xDG
s.t.
node A: xAC + xAD = 60
node B: xBC + xBD = 9
node C: xAC + xBC = xCE + xCF + xCG
node D: xAD + xBD = xDE + xDF + xDG
node E: xCE + xDE = 20
node F: xCF + xDF = 45
node G: xCG + xDG = 35
xij  0  i, j 9 4 11 6 1 2 5 8 3 C D F E G 60 40 35 45 20 A B 7 8

9. Formulating a Transshipment Problem as a Transportation Problem
motivation: simple solution method for transportation problems
how to transform
is node C (D) a source (i.e., a supplier)? a sink (i.e., a customer)? 9 4 11 6 1 2 5 8 3 C D F E G 60 40 35 45 20 A B 7 9

10. Formulating a Transshipment Problem as a Transportation Problem
nodes C and D: both a source and a sink 9 4 11 6 1 2 5 8 3 C D F E G 60 40 35 45 20 A B 7
two linked transportation problems 9 4 11 C D 60 40 A B 7 6 1 2 5 8 3 C’ F E G 35 45 20 D’ 10

11. Formulating a Transshipment Problem as a Transportation Problem
unsure flows: 0  xCC’  100, 0  xDD’  100
sink C D E F G
source A 7 4  60 B 9 11 40 ? 3 5 2 1 8 6 20 45 35 11

12. Formulating a Transshipment Problem as a Transportation Problem
observation: internal flow of zero cost does not affect the problem
flow of 20 (or 2,000) units at no cost from node C to node C does not change the problem 9 4 11 6 1 2 5 8 3 C D F E G 60 40 35 45 20 A B 7 12

13. Formulating a Transshipment Problem as a Transportation Problem
unsure flows: 0  xCC’  100, 0  xDD’  100
sink C D E F G
source A 7 4  60 B 9 11 40 3 5 2
100 1 8 6 20 45 35
100 9 4 11 6 1 2 5 8 3 C D F E G 60 40 35 45 20 A B 7 13

14. Formulating a Transshipment Problem as a Transportation Problem
unsure flows: 0  xCC’  100, 0  xDD’  100 7
100 9 4 11 6 1 2 5 8 3 C D F E G 60 40 35 45 20 A B
100 9 4 11 6 1 2 5 8 3 C D F E G 60 40 35 45 20 A B 7 14

15. Formulating a Transshipment Problem as a Transportation Problem
interpretation of the flow pattern, e.g., 10 25 90 60
100 30 45 20 C D F E G 40 35 A B 10 30 25 45 20 C D F E G 60 40 35 A B 15

16. Capacitated Transshipment Problems
lower and upper bounds for xij
0 ≤ lij ≤ xij ≤ uij
any algorithms solving transshipment problems can solve the capacitated version of a transshipment problem 16

17. Exercise Model the problem as a balanced transportation problem 17 2 94 11 6 1 5 8 3 C D F E G 60 40 35 45 20 A B 7 17

18. Minimum Cost Flow Problems18

19. Minimum Cost Flow Problems
A = the set of assignment problems
T = the set of transportation problems
TS = the set of transshsipment problems
MCF = the set of minimum cost flow problems
A  T  TS  MCF 19

20. Minimum Cost Flow Problems
balanced flow
directed arcs
an undirected arc replaced by two directed arcs with opposite directions 20

21. Example 5.4 of [7] 21

22. based on conservation of flow
Example 5.4 of [7]
min 5x02+4x13+2x23+6x24+5x25+x34+2x37
+4x42+6x45+3x46+4x76,
s.t.
a constraint for a node,
based on conservation of flow 22

23. Minimum Cost Flow Problems
special structure
optimal integral solution if all availabilities, requirements, and capacities being integral
solution methods: linear programming (i.e., Simplex), transportation Simplex, network flow methods 23

24. Minimum Cost Flow Problems with Bounds
two general approaches to solve lij  xij  uij
either algorithms specially for bounded MCF problems
or converting a bounded MCF problem to an unbounded one 24

25. Converting a Bounded MCF to an Unbounded One
what does the paragraph mean?
if you don’t know what to do, work with a simple numerical example. 25

26. Minimum Cost Flow c01 = 3, c02 = 1, c12 = 2 inflow of node 0 = 8
c01 = 3, c02 = 1, c12 = 2
inflow of node 0 = 8
outflow of node 2 = 8
MCF = ?
all 8 units through (0, 2), of cost 8 26

27. Relaxing a Lower Bound MCF: all 8 units through (0, 2), of cost 8
suppose 5  x01
how to convert the problem into an unbounded MCF problem? 8 3 1 2 27

28. Relaxing a Lower Bound 8 3 1 2
LP formulation 28

29. Relaxing a Lower Bound 8 3 1 2
define y01 = x01-5  29

30. How Does the Network Look Like?5 8 3 1 2
an unbounded network 30

31. Relaxing the Lower Bound l01 = 58 3 1 2  8 3 1 2
objective function changed to min 3y01+2x12+x02+15
question: is it possible to convert to a network of the objective function without adding 15 by oneself? 31

32. Relaxing the Lower Bound l01 = 58 3 1 2  8 3 1 2  5 8 3 1 2 a
how about
adding a dummy node a such that
c0a = 3, ca1 = 0, outflow from a = 5
adding a dummy inflow of 5 to node 1 32

33. Relaxing an Upper Bound
MCF: all 8 units through (0, 2), of cost 8
suppose x02  7
how to convert the problem into an unbounded MCF problem? 8 3 1 2 33

34. Relaxing an Upper Bound
suppose x02  7 8 3 1 2 8 3 1 2 a 7 34

35. Negative Cost?
possible to have negative cost as long as there is no negative cost cycle
e.g., if c24 = -6 35

36. Generalization of MCF
from one commodity (i.e., product) to multi-commodity (i.e., multiple products)
flow without gain to with gain 36

37. Multi-Commodity Flow Problems
two types of products
(cij, uij):
(cost, upper bound) of an arc
type 2: 3
(3, 4)
(1, 6)
(2, 4) 1 2
type 1: 5 37

38. Multi-Commodity Flow Problems
“An extension of the problem of finding the minimum cost flow of a single commodity through a network is the problem of minimizing the cost of the flows of several commodities through a network. This is the minimum cost multi-commodity network flow problem. There will be capacity limitations on the flows of individual commodities through certain arcs as well as capacity limitations on the total flow of all commodities through individual arcs.” …
“The resultant model has a block angular structure of the type discussed in Section 4.1.” 38

39. LP Formulation of a Two-Commodity Flow Problem
let xij be the flow of type 1 commodity along arc (i, j)
let yij be the flow of type 1 commodity along arc (i, j)
type 2: 3
(3, 4)
(1, 6)
(2, 4) 1 2
type 1: 5 39

40. LP Formulation of a Two-Commodity Flow Problem
type 1: 5
type 2: 3
(3, 4)
(1, 6)
(2, 4) 1 2
“The resultant model has a block angular structure of the type discussed in Section 4.1.” 40

41. LP Formulation of a Two-Commodity Flow Problem
type 1: 5
type 2: 3
(3, 4)
(1, 6)
(2, 4) 1 2
objective function
flow conservation
of type 1 product
of type 2 product
arc capacity constraints 41

42. Network Flow with Gains Model
flows not conserved
x units into arc (i, j), ijx units out of node j, ij  1
solved by integer programming if integral values required 42

43. Problem of [7] 43