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05. Networks 2 (OR Models)

# 05. Networks 2 (OR Models) - ProgrammingModels Topics...

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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
Min-Cost Flow 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

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Notation for Min-Cost Flow Problem 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   )
Spreadsheet Input Data 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 i j u ij c ij g ij external flow s i  or - d i

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Data Entry Using Math Programming/Network Add-in.  And here is the solution ... Network Model Name: dist_1 Solver: Jensen Network Ph. 1 Iter. 13 TRUE Type: Net Type: Linear Total Iter. 15 FALSE Change Goal: Min Sens.: No Comp. Time 00:00 TRUE Objective: 5300 Side: No Status Optimal FALSE Solve Select the Relink Buttons command from the OR_MM menu before clicking a button. FALSE 100 Arc Data and Flows Node Data and Balance Constraints 100 Num. Name Flow Origin Term. Upper Cost Num. Name Fixed Balance 0 1 Phoe-Chi 200 1 2 200 6 1 Phoe 700 0 60 2 Phoe-LA 200 1 3 200 3 2 Chi -200 0 FALSE 3 Phoe-Dal 200 1 4 200 3 3 LA -200 0 FALSE 4 Phoe-Atl 100 1 5 200 7 4 Dal -300 0 FALSE 5 Dal-LA 0 4 3 200 5 5 Atl -150 0 6 Dal-Chi 0 4 2 200 4 6 NY -250 0 7 Dal-NY 50 4 6 200 6 7 Aus 200 0 8 Dal-Atl 50 4 5 200 2 8 Gain 200 0 9 Atl-NY 0 5 6 200 5 10 Atl-Dal 0 5 4 200 2 11 Atl-Chi 0 5 2 200 4 12 Aus-LA 0 7 3 200 7 13 Aus-Dal 200 7 4 200 2 14 Aus-Atl 0 7 5
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05. Networks 2 (OR Models) - ProgrammingModels Topics...

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