19generalizedflows

# 19generalizedflows - 15.082J and 6.855J Generalized Flows 1...

This preview shows pages 1–11. Sign up to view the full content.

1 15.082J and 6.855J Generalized Flows

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 Overview of Generalized Flows Suppose one unit of flow is sent in (i,j). We relax the assumption that one unit arrives at node j. If 1 unit is sent from i, µ ij units arrive at j. µ ij is called the multiplier of (i,j) i j µ ij = 7 We will present: ± LP Formulation ± Two applications ± Generalized Network Simplex Algorithm
3 LP Formulation of Generalized Flows x ij = amount of flow sent in (i,j) µ ij = multiplier of (i,j) b(i) = supply at node i c ij = unit cost of flow in (i,j) u ij = upper bound on flow in (i,j) (,) Minimize ij ij ij A cx :( , ) ) subject to ( ) ij ji ji jij A x xb i µ ∈∈ = 0 f o r a l l ( , ) . ij ij xu i jA ≤∈

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
4 Conversions of physical entities i j 4/1/03 4/1/04 µ ij = 1.05 (i,j) represents a 1 year investment in a CD. i j coal electricity µ ij = .4 (i,j) represents a conversion of coal into electricity
5 Machine Scheduling i j job machine µ ij = 3 It takes 3 hours to make one unit of job i on machine j. x ij = proportion of product i made on machine j µ ij = number of hours to make product i on machine j d(i) = number of units of product i that need to be made. The total time available on machine j is u j

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
6 Flows Along Directed Paths Suppose that 1 unit is sent from node 1, that flow is conserved in 2, 3, and 4, arrives at node 5. 1 µ 12 = 3 2 µ 23 = .5 3 µ 34 = 4 4 µ 45 = 1 5 1 3 1.5 6 6 For a directed path P from i to j, if one unit of flow is sent from i, then the amount arriving at j is: (,) () ij ij P P µ =
7 Flows Along Non-directed Paths Suppose that 1 unit is sent from node 1, that flow is conserved in 2, 3, and 4, arrives at node 5. 1 µ 12 = 4 2 µ 23 = 2 3 µ 34 = 6 4 µ 45 = 4 5 1 4 2 12 3 1 -2 2 -3 Let P be a path from i to j. Forward arcs of P P = Backward arcs of P P = If one unit of flow is sent from i, then the amount arriving at j is: (,) () / ij ij ij P P µ µµ ∈∈ =

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
8 Flows Along Cycles 1 4 2 12 3 1.5 (,) () / ij ij ij W W µ µµ ∈∈ = ∏∏ 1 4 2 2 3 6 4 4 5 2 W Suppose 1 unit is sent around W starting and ending at node 1. 1 -2 -3 2 -1.5 µ (W) = 1.5 If µ (W) 1, then the amount of flow arriving at node 1 is different then the amount leaving node 1. If µ (W) = 1, W is called a breakeven cycle .
9 Flows Along Cycles 1 4 2 12 3 1.5 s 4 2 2 3 6 4 4 5 2 W Suppose θ units are send around W starting and ending at node s. The net amount arriving at node 1 is: θ [ µ (W)- 1 ]. To create a “supply” of α at node s, send α/ [ µ (W)- 1 ] units of flow.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
10 On the LP for Generalized Flows (,) Minimize ij ij ij A cx :( , ) ) subject to ( ) ij ji ji jij A x xb i µ ∈∈ = 0 f o r a l l ( , ) . ij ij xu i jA ≤∈ The equality constraints have full row rank, which is n.
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 03/15/2010 for the course IE 505 taught by Professor Yok during the Spring '10 term at Galatasaray Üniversitesi.

### Page1 / 46

19generalizedflows - 15.082J and 6.855J Generalized Flows 1...

This preview shows document pages 1 - 11. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online