# 6417c12 - Minimum Cost Flows CONTENTS Introduction to Minimum Cost Flows(Section 9.1 Applications of Minimum Cost Flows(Section 9.2 Structure of

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1 Minimum Cost Flows CONTENTS Introduction to Minimum Cost Flows (Section 9.1) Applications of Minimum Cost Flows (Section 9.2) Structure of the Basis (Section 11.11) Optimality Conditions (Section 9.3) Obtaining Primal and Dual Solutions (Section 11.4) Network Simplex Algorithm (Section 11.5) Strongly Feasible Basis (Section 11.6)

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2 Minimum Cost Flow Problem Determine a least cost shipment of a commodity through a network in order to satisfy demands at certain nodes from available supplies at other nodes. Arcs have capacities and cost associated with them. Distribution of products Flow of items in a production line Routing of cars through street networks Routing of telephone calls 1 2 3 4 5 6 7 10 10 -5 -15 5 3 6 4 3 1 2 2 4 6 5
3 Minimum Cost Flow Problem (contd.) Mathematical Formulation : Minimize c x subject to b i for each node i N x u for each arc i j N ij ij i j A ij ij ( , ) ( ) ( , ) - = x x ij ji {j:(j,i) A} {j:(i,j) A} 0 where Supply nodes (b(i) > 0); Demand nodes (b(i) < 0), Transhipment nodes (b(i) = 0) Mass balance constraints (flow in - flow out = supply/demand) Flow bound constraints (lower and upper bound constraints) b i i N ( ) . = 0

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4 Assumptions All data (cost, supply/demand, capacity) are integral. The network is directed. The sum of supplies equals the sum of demands, and the problem admits a feasible flow. The network contains an uncapacitated directed path between every pair of nodes. All arc costs are non-negative.
5 Distribution Problems A car manufacturer has several manufacturing plants and produces several car models at each plant that is shipped to various retailers. The manufacturer must determine the production plan for each model and shipping pattern that minimizes the overall cost of production and transportation. r 1 r 2 p 1 p 2 p 1 /m 1 p 1 /m 2 p 2 /m 1 p 2 /m 2 p 2 /m 3 r 1 /m 1 r 1 /m 2 r 1 /m 3 r 2 /m 1 r 2 /m 2 Plant nodes Plant/model nodes Retailer/model nodes Retailer nodes

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6 Airplane Hopping Problem A hopping flight visits the cities 1, 2, 3, … , n, in a fixed sequence. The plane can pickup passengers at any node and drop them off at any other node. Let b ij denote the number of passengers available at node i who want to go to node j and let f ij denote the fare for such passengers.
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## This note was uploaded on 08/27/2011 for the course ESI 6417 taught by Professor Siriphonglawphongpanich during the Spring '07 term at University of Florida.

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6417c12 - Minimum Cost Flows CONTENTS Introduction to Minimum Cost Flows(Section 9.1 Applications of Minimum Cost Flows(Section 9.2 Structure of

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