Introduction to Operations Research

1
Introduction to Operations Research

• Networks and the Shortest Path Problem

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Networks are everywhere

• Physical Networks
• Road Networks
• Railway Networks
• Airline traffic Networks
• Electrical networks, e.g., the power grid
• Abstract networks
• organizational charts
• precedence relationships in projects
• Others?

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overview

• Networks and graphs are powerful modeling tools.
• Most OR models have networks or graphs as a major
  aspect
• Each representation has its advantages
• Major purpose of a representation
• efficiency in algorithms
• ease of use

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description

• Many important optimization problems can be
  analyzed by means of graphical or network
  representation. In this chapter the following
  network models will be discussed
• Shortest path problems
• Maximum flow problems
• CPM-PERT project scheduling models
• Minimum Cost Network Flow Problems
• Minimum spanning tree problems

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Shortest path problem

• Most efficient way to go from one point to
  another in a distance network or networks
  representing non-distance phenomenon, e.g., the
  cost network representing production,
  inventory, and other costs

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definitions

• Networks consists of a set of points and a set
  of lines connecting certain points.
• Nodes The points in the network
• Arcs The lines in the network
• Directed Arc An arc for which flow is allowed
  in only one direction
• Undirected Arc An arc for which flow is allowed
  in both directions
• Directed Network A network with only directed
  arcs
• Undirected Network A network with only
  undirected arcs
• Directed Path A sequence of arcs from node i to
  node j such that all arcs are directed towards
  node j
• Undirected Path A sequence of arcs from node i
  to node j such that all arcs can be directed
  either towards or away from node j

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definitions

• Cycle A path that begins and ends at the same
  node
• Connected Two nodes with at least one
  undirected path between them
• Connected Network A network where every pair of
  nodes is connected
• Tree A network without any cycles and with the
  number of connected nodes being greater than the
  number of arcs
• Spanning Tree A tree that is a connected
  network
• Arc Capacity The maximum flow that can be
  carried on a directed arc
• Supply Node Flow out of the node exceeds flow
  into the node
• Demand Node Flow into the node exceeds flow out
  of the node
• Transshipment Node Flow into the node equals
  flow out of the node

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Basic Definitions

• A minimum spanning tree is a spanning tree with
  minimum weight.

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The Shortest-Path Problem

• Consider an undirected and connected network with
  two special nodes called the origin and the
  destination.
• Associated with each of the links (undirected
  arcs) is a nonnegative distance. The objective
  is to find the shortest path (the path with the
  minimum total distance) from the origin to the
  destination.

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Shortest Path Problem Algorithm

• Objective of nth iteration Find the nth nearest
  node to the origin (to be repeated for n 1,2, .
  . . until the nth nearest node is the
  destination.)
• Input for nth iteration n - 1 nearest nodes to
  the origin (solved for at the previous
  iterations), including their shortest path and
  distance from the origin. (These nodes, plus the
  origin, will be called solved nodes the others
  are unsolved nodes.)

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Shortest Path Problem Algorithm

• Candidates for nth nearest node Each solved node
  that is directly connected by a link to one or
  more unsolved nodes provides one candidate-the
  unsolved node with the shortest connecting link.
  (Ties provide additional candidates.)
• Calculation of nth nearest node For each such
  solved node and its candidate, add the distance
  between them and the distance of the shortest
  path from the origin to this solved node. The
  candidate with the smallest such total distance
  is the nearest node (ties provide additional
  solved nodes), and its shortest path is the one
  generating this distance.

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Seervada Park

• Cars are not allowed into the park
• There is a narrow winding road system for trams
  and for jeeps driven by the park rangers
• The road system is shown without curves in the
  next slide
• Location O is the entrance into the park
• Other letters designate the locations of the
  ranger stations
• The scenic wonder is at location T
• The numbers give the distance of these winding
  roads in miles
• The park management wishes to determine which
  route from the park entrance to station T has the
  smallest total distance for the operation of the
  trams

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Shortest path

• Consider an undirected and connected network
  with origin and destination nodes. Associated
  with every arc is a non-negative distance. The
  objective is to find the shortest path from the
  origin to the destination.

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A
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Shortest path

• 1st iteration Step 1 Neighboring Nodes A,
  B, C
• Step 2 Shortest path from O to neighboring
  nodes Min 2, 5, 4 2
• Step 3 The shortest path from O to A S
  O, A

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Shortest path

• 2nd iteration Step 1 Neighboring nodes B,
  C, D
• Step 2 Min (Min (2 2, 5), 4, (2 7)) 4.
• Step 3 Shortest path B to C S O, A, B, C
  

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Shortest path

• 3rd iteration Step 1 Neighboring nodes D,
  E. Only AD, BD, BE, and CE
• Step 2 Min(Min(2 7, 44), Min(4 3, 44))
  7
• Step 3 The shortest path to E S O, A, B,
  C, E

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Shortest path

• 4th iteration Step 1 Include Nodes D and T.
  Include arcs AD, BD, ED, ET
• Step 2 Min((min(27, 44, 71), (77))) 8
• Step 3 Shortest path from node O to D S
  O, A, B, C, E, D

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