Chapter 10 Network Flows and Graphs q Network

10. 1 Graphs, Networks, and Flows q Opt Theory 2019 2

q Cost tables From/To 3: Memphis 4: Pittsburgh 1: Wisconsin 7 8 2: Alabama

FIGURE 10. 1 Network for Optimal Ovens, Incorporated (OOI) Application Optimization in Operations Research,

FIGURE 10. 2 Minimum Cost Network Flow Problem for OOI Application Optimization in Operations

FIGURE 10. 3 Optimal Flows for OOI Application Optimization in Operations Research, 2 e

FIGURE 10. 4 Starting Phase I Flow for the OOI Application Optimization in Operations

q Time-Expanded Flow Models and Networks q many applications of network flows involve time-expanded

FIGURE 10. 5 Agrico Time-Expanded Network Format Optimization in Operations Research, 2 e Ronald

TABLE 10. 1 Node–Arc Incidence Matrix of the OOI Application Optimization in Operations Research,

10. 2 Cycle Directions for Network Flow Search q Want to identify improving feasible

FIGURE 10. 6 Chains and Cycles of the OOI Network continued on next slide

FIGURE 10. 6 (continued) Chains and Cycles of the OOI Network continued on next

FIGURE 10. 6 (continued) Chains and Cycles of the OOI Network +1 +1 -1

FIGURE 10. 6 (continued) Chains and Cycles of the OOI Network Optimization in Operations

FIGURE 10. 7 Possible Ways a Cycle Can Visit a Node Optimization in Operations

FIGURE 10. 8 Initial Flow x(0) in OOI Application (4, ∞) (17, ∞) (5,

FIGURE 10. 9 An Improving Feasible Cycle Direction at x(0) of Figure 10. 8

FIGURE 10. 10 A Non-Cycle Improving Feasible Direction at x(0) of Figure 10. 8

q Rudimentary Cycle Direction Search of the OOI Application q Figure 10. 12 Ø

FIGURE 10. 11 Data and Initial Flow for the OOI Application Optimization in Operations

FIGURE 10. 12 Rudimentary Cycle Direction Solution of OOI Application continued on next slide

FIGURE 10. 12 (continued) Rudimentary Cycle Direction Solution of OOI Application continued on next

FIGURE 10. 12 (continued) Rudimentary Cycle Direction Solution of OOI Application Optimization in Operations

10. 3 Cycle Cancelling Algorithms for Optimal Flows q Opt Theory 2019 37

FIGURE 10. 13 Residual Digraph for OOI Application Flow of Figure 10. 12 Optimization

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Chapter 10. Network Flows and Graphs q Network flow problems are special, yet widely applicable cases of linear programs which prove even more tractable. Much larger models can be solved, because specialized algorithms apply. Most important, discrete cases, which we know are usually more difficult, can often be managed with no extra effort at all. Opt Theory 2019 1

10. 1 Graphs, Networks, and Flows q Opt Theory 2019 2

q Cost tables From/To 3: Memphis 4: Pittsburgh 1: Wisconsin 7 8 2: Alabama 4 7 From/To 5: Fresno 6: Peoria 7: Newark 3: Memphis 25 5 17 4: Pittsburgh 29 8 16 Opt Theory 2019 3

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q Time-Expanded Flow Models and Networks q many applications of network flows involve time-expanded formulations to account for flows over time. This is especially true for those involving inventory management. (also for scheduling problems in airline industry, railroad, …) In the specially structured network flow case, such problems lead to time expanded networks. q Def 10. 7: Time-expanded networks model each node of a flow system as a series of nodes, one for each time interval. Arcs than reflect either flows between points in a particular time, or flows across time in a particular location. q Application 10. 2: Agrico Chemical Time-Expanded Network Flow (Fertilizer production and distribution, and inventory) Products for this real company originate at 4 plants and are transshipped through 20 regional distribution centers before reaching customers in any of 500 service areas. Agrico's decision problem requires choosing the amount to produce in each of the four quarters of the year, the pattern of shipping and storing it at distribution sites, and a scheme for sending it on to customers. We want to do all this at minimum total cost. Opt Theory 2019 12

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10. 2 Cycle Directions for Network Flow Search q Want to identify improving feasible directions for the network algorithm. q Chains, Paths, Cycles, and Dicycles q Def 10. 9: A chain is a sequence of arcs connecting two nodes. Each arc has exactly one node in common with its predecessor in the sequence, and no node is visited more than once (arc directions not relevant). q Def 10. 10: A cycle is a chain with the same beginning and ending node. q Def 10. 11: Paths are chains that transmit all arcs in the forward direction. q Def 10. 12: Dicycles are cycles that have all arcs oriented in the same direction. Opt Theory 2019 16

FIGURE 10. 6 Chains and Cycles of the OOI Network continued on next slide Optimization in Operations Research, 2 e Ronald L. Rardin Copyright © 2017, 1998 by Pearson Education, Inc. All Rights Reserved

FIGURE 10. 6 (continued) Chains and Cycles of the OOI Network continued on next slide Optimization in Operations Research, 2 e Ronald L. Rardin Copyright © 2017, 1998 by Pearson Education, Inc. All Rights Reserved

FIGURE 10. 6 (continued) Chains and Cycles of the OOI Network +1 +1 -1 -1 continued on next slide Optimization in Operations Research, 2 e Ronald L. Rardin Copyright © 2017, 1998 by Pearson Education, Inc. All Rights Reserved

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FIGURE 10. 9 An Improving Feasible Cycle Direction at x(0) of Figure 10. 8 (4, ∞) (17, ∞) (5, ∞) Optimization in Operations Research, 2 e Ronald L. Rardin Copyright © 2017, 1998 by Pearson Education, Inc. All Rights Reserved

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FIGURE 10. 10 A Non-Cycle Improving Feasible Direction at x(0) of Figure 10. 8 Optimization in Operations Research, 2 e Ronald L. Rardin Copyright © 2017, 1998 by Pearson Education, Inc. All Rights Reserved

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q Rudimentary Cycle Direction Search of the OOI Application q Figure 10. 12 Ø t = 0: cycle 2 -4 -7 -3 -2 Ø t = 1: cycle 2 -3 -1 -4 -2 Ø t = 2: cycle 2 -3 -4 -2 Opt Theory 2019 31

FIGURE 10. 12 Rudimentary Cycle Direction Solution of OOI Application continued on next slide Optimization in Operations Research, 2 e Ronald L. Rardin Copyright © 2017, 1998 by Pearson Education, Inc. All Rights Reserved

FIGURE 10. 12 (continued) Rudimentary Cycle Direction Solution of OOI Application continued on next slide Optimization in Operations Research, 2 e Ronald L. Rardin Copyright © 2017, 1998 by Pearson Education, Inc. All Rights Reserved

FIGURE 10. 12 (continued) Rudimentary Cycle Direction Solution of OOI Application Optimization in Operations Research, 2 e Ronald L. Rardin Copyright © 2017, 1998 by Pearson Education, Inc. All Rights Reserved

10. 3 Cycle Cancelling Algorithms for Optimal Flows q Opt Theory 2019 37