Operations Research Class Notes Network Models

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Operations Research Class NotesNetwork Models

• Yanni Papadakis

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Network Model Types

• Transportation Problem
• Transshipment Problem
• Assignment Problem
• Min Cost Flow Problem Family
• Max Flow Problem
• Shortest Path Problem
• Min Spanning Tree Problem

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Transportation Problem
Parameters Capacity Limit Transportation Cost
c(i,j)factory i to retailer j Demand Limit
C1 C2 C3
FACTORIES
RETAILERS
D1 D2 D3 D4
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Transportation Problem Structure

• Factories I3
• Retailers J4
• Decision Variables IJ12
• Quantity to move from each Factory to each
  Retailer
• Demand Constraints (gt Limit)
• Capacity Constraints (lt Limit)
• Objective Min Total Cost
• Note If Parameters Integer then Solution is
  Integer. No need for IP (unimodularity property)

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Transshipment Problem
Parameters Capacity Limit Transportation Cost
c(i,j)node i to node j Demand Limit
C1 C2 C3
FACTORIES
CROSSDOCK
RETAILERS
D1 D2 D3 D4
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Transshipment Problem Structure

• Factories I3
• Retailers J4
• Crossdock Nodes K2
• Decision Variables IKKJ14
• Quantity/Flow to move between nodes
• Demand Constraints (gt Limit)
• Capacity Constraints (lt Limit)
• Flow Conservation in Crossdocks
• What goes in equals what comes out (No gain or
  Loss)
• Objective Min Total Cost

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Assignment Problem
Parameters Capacity Limit (often equal to
1) Job Cost c(i,j)worker i to job
j Demand Limit(often equal to 1)
C1 C2 C3
WORKERS
JOBS
D1 D2 D3
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Assignment Problem Structure

• Workers I3
• Jobs J3
• Decision Variables IJ9
• Assignment Index worker i to job j
• Job Constraints (gt Limit)
• Worker Constraints (lt Limit)
• Objective Min Total Cost

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Min Cost Flow Problem Family
Parameters Cost c(i,j)node i to node
j Node Capacity Limit (Upper or Lower) Arc
Capacity Limit (Upper or Lower)
C1 C2 C3
SOURCES
NODES
SINKS
D6 D7 D8 D9
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Min Cost Flow Problem Structure

• Nodes (Sources, Sinks, Distribution Nodes) total
  I3249
• Arcs total 14
• Flow f(i,j) on arc connecting nodes i and j
• Node Capacity Limit L(i)
• Sum f(j,i)ltL(i) Source Capacity
• Sum f(j,i)gtL(i) Sink Demand
• Sum f(j,i) 0 Conservation of Flow
• Arc Capacity Limit b(i,j) gt (or lt) f(i,j)
• Objective Min Total Cost
• Note 1 Integrality Property Holds
• Note 2 Assume No Gain (as with interest on
  investments) or Loss (as in Electrical networks)

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Max Flow Problem
Parameters Arc Capacity Upper Limit
C1 C2 C3
SOURCES
NODES
SINKS
D6 D7 D8 D9
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Max Flow Problem Structure

• Belongs to Min Cost Flow Problem
• Objective Max Flow Out of Sinks
• Constraints
• Flow Conservation
• Arc Capacity
• Note Very efficient algorithms exist to solve
  this problem. But we use generic methods. Why?
• Additional Constraints that violate Max-Flow
  structure could be required in practice
• Use special software only in big problems solved
  repetitively

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Shortest Path Problem
D
A
F
C
G
B
E
Shortest Distance from A to G
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Shortest Path Problem Structure

• Belongs to Min Cost Flow Problem
• Objective Min Total Cost
• Cost is Distance
• Constraints
• Flow Conservation
• Sink Out Flow is 1
• Source Out Flow in -1
• Note Very efficient algorithms exist to solve
  this problem. But we use generic methods. Why?
• Additional Constraints that violate Max-Flow
  structure could be required in practice
• Use special software only in big problems solved
  repetitively

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Min Spanning Tree Problem
D
A
F
C
G
B
E
Connect All Points at Least Cost SOLVE USING
GREEDY ALGORITHM