The Minimum Cost Network Flow Problem

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1 EMIS 8374 [MCNFP Review] 1 The Minimum Cost Network Flow Problem Problem Instance: Given a network G = (N, A), with a cost c ij, upper bound u ij, and lower bound l ij associated with each directed arc (i, j), and supply of, or demand for, b(i) units of some commodity at each node. Supply Nodes: b(i) > 0 Transshipment Nodes: b(i) = 0 Demand Nodes: b(i) < 0

2 EMIS 8374 [MCNFP Review] 2 Linear Programming Formulation: that of finding a minimum-cost feasible flow: The problem is min subject to (i,j) A {j:(i,j) A} c ij x ij x ij {j:(j,i) A} l ij x ij u ij (i, j) A x ji = b(i) i N, Note the problem is not feasible unless the supplies and demands are balanced (i.e., i N b(i) = 0).

3 EMIS 8374 [MCNFP Review] 3 The Circulation Problem: A MCNFP where b(i) = 0 for all nodes. The Transportation Problem: A MCNFP where A bipartite network where N = N 1 N 2 (N 1 N 2 = ) N 1 are supply nodes and N 2 are demand nodes i N 1 and j N 2 for all (i, j) A The Assignment Problem: A Transportation Problem where N 1 = N 2 b(i) = +1 for all i N 1 and b(j) = 1 for all j N 2

4 EMIS 8374 [MCNFP Review] 4 The Shortest Path Problem: defined on a network with arc costs, but no capacity limits. The objective is to find a path from node s, the source, to node t, the sink, that minimizes the sum of the arc costs along the path. To formulate as MCNFP: s is a suppy node with b(s) = 1. t is a demand node with b(t) = 1. All other nodes (N \ {s, t}) are transshipment nodes. l ij = 0 and u ij = 1 for all arcs.

5 EMIS 8374 [MCNFP Review] 5 LP Formulation of the Shortest Path Problem min subject to (i,j) A c ij x ij {j:(s,j) A} {j:(i,j) A} {j:(t,j) A} x sj x ij x tj {j:(j,s) A} {j:(j,i) A} {j:(j,t) A} x js = 1, 0 x ij 1 (i, j) A. x ji = 0 i N \ {s, t}, x jt = 1,

6 EMIS 8374 [MCNFP Review] 6 The Maximum Flow Problem: defined on a directed network with capacities, and no costs. In addition two nodes are specified, a source node, s, and sink node, t. The objective is the find the maximum possible flow between the source and sink while satisfying the arc capacities.

7 EMIS 8374 [MCNFP Review] 7 LP Formulation of the Maximum Flow Problem max subject to v {j:(s,j) A} {j:(i,j) A} {j:(t,j) A} x sj x ij x tj {j:(j,s) A} {j:(j,i) A} {j:(j,t) A} x js = v, x ji = 0 i N \ {s, t}, x jt = v, l ij x ij u ij (i, j) A. The scalar variable v is referred to as the value of the flow vector x = {x ij }.

8 EMIS 8374 [MCNFP Review] 8 The Max Flow Problem Formulated as MCNFP Convert the problem to an equivalent minimum cost circulation problem as follows: Let c ij = 0 for all (i, j) A. Let b(i) = 0 for all i N. Add an arc from s to t with cost c st = 1. min x ts s.t. {j:(i,j) A} x ij {j:(j,i) A} l ij x ij u ij (i, j) A. x ji = 0 i N,

9 EMIS 8374 [MCNFP Review] 9 The Texas Confectionery Company (TCC) produces three types of candy bars at two different plants. Houston plant produces Rice Krunchy and Aggie bars. Austin plant produces Aggie Bars and Longhorn Bars. There are 160 hours of production time available per month at each plant.

10 EMIS 8374 [MCNFP Review] 10 Product Production Cost Product Time Demand Name Houston Austin (Minutes) (Units) Rice Krunchy Bar $ Aggie Bar $0.05 $ Longhorn Bar - $ (a) Formulate a MCNFP that TCC can solve to determine how to minimize the cost of meeting the demand for its products. Briefly describe what the elements (nodes, arcs, capacities, etc.) of your network model represent. Hint: How many hours of production time are required to meet the demand for each type of candy bar?

11 EMIS 8374 [MCNFP Review] 11 (a) Nodes: H and A are supply nodes representing the Houston and Austin plants, respectively. These nodes each have a supply of 9600 minutes of production time. RK, AB and LB are demand nodes representing the demand for Rice Krunchy, Aggie and Longhorn Bars, respectively. The demand at RK is 6000 minutes (30 minutes times 200 units), the demands at AB and LB are 5600 and 4800 minutes, respectively. D is a dummy node to absorb the excess supply of minutes.

12 EMIS 8374 [MCNFP Review] 12 Arcs: The arc costs for (H, RK), (H, AB), (A, AB) and (A, LB) are $ , $ , $ and $ , respectively. These arcs are uncapacitated. A unit of flow on arc (i, j) represents one minute of production time at plant i used for making bar j. The arcs from the supply nodes to the dummy node have zero cost, zero lower bound and an upper bound of 9600.

13 EMIS 8374 [MCNFP Review] 13 Network for TCC Problem 9600 H RK D AB A LB -4800

14 EMIS 8374 [MCNFP Review] 14 (b) Suppose that TCC wants to plan two months in advance and believes that it will need 240 Rice Krunchy Bars, 360 Aggie Bars and 480 Longhorn bars at the end of next month. If there is time left over in the current month (after the production for this month s demand is finished), then some of these bars can be manufactured this month and held in inventory until they are needed. Extend your MCNFP from part (a) to minimize TCC s cost for meeting the demand for its products over the next two months.

15 EMIS 8374 [MCNFP Review] 15 (b) Rename nodes RK, AB and LB to RK1, AB1 and LB1 (demand in month 1) and add nodes RK2, AB2 and LB2 to represent the demand in month 2. Add nodes RKI ABI and LBI to represent the surplus production of Rice Krunchy, Aggie and Longhorn bars from month 1 that is held inventory. Rename nodes H and A to H1 and A1, respectively, and add nodes H2 and A2 to represent production in month two.

16 EMIS 8374 [MCNFP Review] H2 RK2 RKI H1 RK D -400 AB2 ABI AB A2 LB2 LBI A1 LB

17 EMIS 8374 [MCNFP Review] 17 Arc Cost Arc Cost Arc Cost $0.04 $0.04 $0.04 (H1, RK1) 30 (H1, RKI) 30 (H2, RK2) 30 $0.05 $0.05 $0.05 (H1, AB1) (H1, ABI) (H2, AB2) $0.06 $0.06 $0.06 (A1, AB1) 20 (A1, ABI) 20 (A2, AB2) 20 $0.06 $0.06 $0.06 (A1, LB1) 15 (A1, LBI) 15 (A2, LB2) 15 All arcs have zero lower bound and can have an infinite upper bound. Arcs not listed above have zero cost. Note that the network can be simplified as shown in the next figure.

18 EMIS 8374 [MCNFP Review] RK2 H2 H1 RK D -400 AB2 AB A2 LB2 A1 LB

19 EMIS 8374 [MCNFP Review] 19 (c) Candy bars that are stored in inventory must be kept in a special storage facility so that they do not become stale. How would you extend your MCNFP from part (b) to account for inventory holding costs of $0.02 for each Rice Krunchy Bar, $0.01 for each Aggie bar and $0.05 for each Longhorn bar?

20 EMIS 8374 [MCNFP Review] 20 (c) Add the storage cost to the cost on the arcs from H1 and A2 to RK2, AB2, and LB2 Arc Cost Arc Cost $0.04+$0.02 $0.05+$0.01 (H1, RK2) (H1, AB2) $0.06+$0.01 $0.06+$0.05 (A1, AB2) 20 (A1, LB2) 15

Graphs and Network Flows IE411. Lecture 12. Dr. Ted Ralphs

Graphs and Network Flows IE411 Lecture 12 Dr. Ted Ralphs IE411 Lecture 12 1 References for Today s Lecture Required reading Sections 21.1 21.2 References AMO Chapter 6 CLRS Sections 26.1 26.2 IE411 Lecture