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# NF2 - ProgrammingModels Topics Min-cost flow...

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Lecture 5 – Integration of Network Flow  Programming Models Topics Min-cost flow problem (general model) Mathematical formulation and problem characteristics Pure vs. generalized networks

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GAINS 8 ATL 5 NY 6 DAL 4 CHIC 2 AUS 7 LA 3 PHOE 1 (6) (3) (5) (7) (4) (2) (4) (5) (5) (6) (4) (7) (6) (3) [–150] [200] [–300] [200] [–200] [–200] (2) (2) (7) [–250] [700] [supply / demand] (shipping cost) arc lower bounds = 0 arc upper bounds = 200 Distribution Problem
• Warehouses store a particular commodity in Phoenix, Austin and  Gainesville.  • Customers - Chicago, LA, Dallas, Atlanta, & New York  Supply [  s i  ] at each warehouse  i Demand [  - d j   ] of each customer  j • Shipping links depicted by arcs, flow on each arc is limited to  200 units. • Dallas and Atlanta - transshipment hubs • Per unit transportation cost ( c ij   ) for each arc  Problem: Determine optimal shipping plan that minimizes  transportation  costs  Example :   Distribution problem Min-Cost Flow Problem

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In general: [supply/demand] on nodes (shipping cost per unit) on arcs In example: all arcs have an upper bound of 200 nodes labeled with a number 1,...,8 • Must indicate notation that is included in model: ( c ij   ) unit flow cost on arc ( i ,   j   ) ( u ij   ) capacity (or simple upper bound) on arc ( i ,   j   ) ( g ij   ) gain or loss on arc ( i ,   j   ) • All 3 could be included: ( c ij   u ij   g ij   ) Notation for Min-Cost Flow Problem
arc name  termination node      cost  gain origin node lower bound upper bound x ij i j l ij The origin node is the arc’s  tail The termination node is called the  head Supplies are  positive  and demands are  negative External flow balance:  total supply = total demand i j u ij c ij g ij external flow s i  or - d i Spreadsheet Input Data

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And here is the solution ... Data Entry Using Jensen Network Solver Network Model Name: Net1 Solver: Excel Solver Ph. 1 Iter. 13 5300 Type: Net Type: Linear Total Iter. 15 17 Change Goal: Min Sens.: Yes Comp. Time 00:06 TRUE Cost: 5300 Side: No Status Optimal TRUE Solve TRUE 100 Vary Arc Data and Flows Node Data and Balance Constraints Num. Name Flow Origin Term. Upper Cost Red. Cost Num. Name Fixed Balance Dual Values Basis 1 Phoe-Chi 200 1 2 200 6 -3 1 Phoe 700 0 -11 -4 2 Phoe-LA 200 1 3 200 3 -7 2 Chi -200 0 -2 6 3 Phoe-Dal 200 1 4 200 3 -2 3 LA -200 0 -1 12 4 Phoe-Atl 100 1 5 200 7 0 4 Dal -300 0 -6 -7 5 Dal-LA 0 4 3 200 5 0 5 Atl -150 0 -4 8 6 Dal-Chi 0 4 2 200 4 0 6 NY -250 0 0 27 7 Dal-NY 50 4 6 200 6 0 7 Aus 200 0 -8 -13 8 Dal-Atl 50 4 5 200 2 0 8 Gain 200 0 -8 -16 9 Atl-NY 0 5 6 200 5 1 10 Atl-Dal 0 5 4 200 2 4 11
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