Lecture 3 Transshipment Problems Minimum Cost Flow Problems

1
Lecture 3 Transshipment ProblemsMinimum Cost
Flow Problems
2
Agenda

• transshipment problems
• minimum cost flow problems

3
Transshipment Problems
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

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 3xCE 5xCF
  2xCG
• xDE 8xDF 6xDG

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

?????????????????????????
7
LP Formulation of Transshipment Problems
min objective of a flow pattern, s.t.
conditions to be a flow pattern.

• 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

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
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)?

10
Formulating a Transshipment Problem as a
Transportation Problem

• nodes C and D both a source and a sink

two linked transportation problems
11
Formulating a Transshipment Problem as a
Transportation Problem

• unsure flows 0 ? xCC ? 100, 0 ? xDD ? 100

sink sink sink sink sink
C D E F G
source A 7 4 ? ? ? 60
source B 9 11 ? ? ? 40
source C ? ? 3 5 2 ?
source D ? ? 1 8 6 ?
? ? 20 45 35
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

13
Formulating a Transshipment Problem as a
Transportation Problem

• unsure flows 0 ? xCC ? 100, 0 ? xDD ? 100

sink sink sink sink sink
C D E F G
source A 7 4 ? ? ? 60
source B 9 11 ? ? ? 40
source C 0 ? 3 5 2 100
source D ? 0 1 8 6 100
100 100 20 45 35
14
Formulating a Transshipment Problem as a
Transportation Problem

• unsure flows 0 ? xCC ? 100, 0 ? xDD ? 100

15
Formulating a Transshipment Problem as a
Transportation Problem

• interpretation of the flow pattern, e.g.,

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

17
Exercise

• Model the problem as a balanced transportation
  problem

18
Minimum Cost Flow Problems
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

20
Minimum Cost Flow Problems

• balanced flow
• directed arcs
• an undirected arc replaced by two directed arcs
  with opposite directions

21
Example 5.4 of 7
22
Example 5.4 of 7

• min 5x024x132x236x245x25x342x37
• 4x426x453x464x76,
• s.t.

a constraint for a node, based on conservation of
flow
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

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

25
Converting a Bounded MCF to an Unbounded One

• what does the paragraph mean?

if you dont know what to do, work with a simple
numerical example.
26
Minimum Cost Flow

• 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

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?

28
Relaxing a Lower Bound

• LP formulation

29
Relaxing a Lower Bound

• define y01 x01-5

?
30
How Does the Network Look Like?

• an unbounded network

31
Relaxing the Lower Bound l01 5
?

• objective function changed to min
  3y012x12x0215
• question is it possible to convert to a network
  of the objective function without adding 15 by
  oneself?

32
Relaxing the Lower Bound l01 5

• 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

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?

34
Relaxing an Upper Bound

• suppose x02 ? 7

35
Negative Cost?

• possible to have negative cost as long as there
  is no negative cost cycle
• e.g., if c24 -6

36
Generalization of MCF

• from one commodity (i.e., product) to
  multi-commodity (i.e., multiple products)
• flow without gain to with gain

37
Multi-Commodity Flow Problems
(cij, uij) (cost, upper bound) of an arc

• two types of products

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.

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)

40
LP Formulation of a Two-Commodity Flow Problem

• The resultant model has a block angular
  structure of the type discussed in Section 4.1.

41
LP Formulation of a Two-Commodity Flow Problem
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

43
Problem 12.19 of 7