Lecture 3 Transshipment Problems Minimum Cost Flow Problems 1

Agenda transshipment problems minimum cost flow problems 2

Transshipment Problems 3

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

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

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

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

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

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

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

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

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

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

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

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

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

Exercise Model the problem as a balanced transportation problem 17 2 9 4 11 6 1 5 8 3 C D F E G 60 40 35 45 20 A B 7 17

Minimum Cost Flow Problems 18

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

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

Example 5.4 of [7] 21

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

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

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

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

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

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

Relaxing a Lower Bound 8 3 1 2 LP formulation 28

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

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

Relaxing the Lower Bound l01 = 5 8 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

Relaxing the Lower Bound l01 = 5 8 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

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

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

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

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

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

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

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

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

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

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

Problem 12.19 of [7] 43