Facilities Planning

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Facilities Planning
Objectives and Agenda 1. Different types of
Facilities Planning Problems 2. Intro to
Graphs as a tool for deterministic optimization
3. Finding the Minimum Spanning Tree (MST) in a
graph 4. Optimum solution of a Facilities
Planning Problem using MST
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Facilities Planning Problems (a) Site Location
Problem
- Where to locate a new/additional facility
Issues Cost, labor availability, wage levels,
govt. subsidies, transportation costs for
materials, taxes, legal issues,
Example New China Oil co. 7 oil wells ? 1
Refinery Where to locate the refinery to minimize
pipeline costs.
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Facilities Planning Problems (b) Site planning
- How many buildings are required at a site,
their locations, sizes, and connections
(materials, data)
Example Athletic Shoe Co. (a) What are the
issues used to determine building locations? (b)
Optimum layout of underground data cables to
connect all buildings?
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Facilities Planning Problems (c) Building
Layout Problem
- Determine the best size and shape of each
department in a building
Example Plastic Mold Co.
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Facilities Planning Problems (d) Department
Layout Problem
- How to layout the machines, work stations, etc.
in a department
Example Old China Bicycle Co. How will you
design the assembly line for assembling 100
bikes/day?
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Facilities Planning Problems
Most Facility Planning Problems have many
constraints ? Mathematical models are very
complex Why do we need to make mathematical
model ?
We will study one (simple) example of the Site
planning Problem
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Example Site Planning Problem
- Join N population centers of a city by Train
System (MTR) - Direct connection lines can be
built between some pairs - Cost of Train network
? total length of lines - Each pair of Stations
must have some train route between them
Example Map of Delhi and some Population centers.
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Example Site Planning Problem
We will use Graphs to solve the example -
Graph theory (in Mathematics) is useful to solve
many problems - We will use one Graph method
Minimum Spanning Trees (MST) - MST can be used
for many different problems
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Introduction and Terminology Graphs
Graph G(V, E), V a set of nodes and E a
set of edges. Each edge links exactly two
nodes, (node1, node2) An edge is incident on
each node on its ends.
Example G(V, E) ( a, b, c, d,
(a, b), (b, c), (b, d), (c, d), (a, d) )
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Graph terminology
Path a sequence of nodes, ltn0, n1, , nk1gt such
that (i) each ni ? V (ii) (ni, ni1) ? E, for
each i 0, .., k Moving on a path traversing
the graph The length of a path number of edges
in the path
Example P lta, b, c, dgt, P 3
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Graph terminology..
Weighted graph each edge ? a real weight
Cycle A closed path ltn0, n1, , nk, n0gt
Directed graph, Digraph each edge has a
direction (tail, head) A directed edge
is incident from the tail, incident to the
head. Tail parent, Head child
Degree of node no. of edges incident on
it Digraph no. of incoming edges
indegree no. of incoming edges outdegree
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Graph terminology
Connected graph a path between every pair of
nodes
unconnected
connected
Strongly connected digraph each node reachable
from every other node
not strongly connected
Strongly connected
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Graph terminology.
Acyclic graph graph with no cycles
A tree is an undirected, acyclic, connected graph
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Example (repeat)
- Join N population centers of a city by Train
System (MTR) - Direct connection lines can be
built between some pairs - Cost of Train network
? total length of lines - Each pair of Stations
must have some train route between them
Example Map of Delhi and some Population centers.
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Minimum spanning Trees Example
Redraw only the graph, with weights ? length of
rail link.
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Properties of optimum solution
Property 1. The optimum set of connections is a
sub-graph M( V, E) of G, such that V V,
and E ? E. Why?
Property 2. The optimum solution is a
tree. Proof (by contradiction) Assume
existence of cycle ltna, nb, , nk, nagt. gt
??
gt Optimum set of railway links is a minimum
spanning tree
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Minimum spanning Trees Prims method
Step 1. Put the entire graph (all nodes and
edges) in a bag. Step 2. Select any one node,
pull it out of the bag (edges incident on this
node will cross the bag) Step 3. Among all
edges crossing the bag, pick the one with MIN
weight. Add this edge to the MST. Step
4. Select the node inside the bag connected to
edge selected in Step 3. Step 5. Pull node
selected in Step 5 out of bag. Step 6. Repeat
steps 3, 4, 5 until the bag is empty.
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Minimum spanning Trees Example
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Minimum spanning Trees Example..
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Minimum spanning Trees Example
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Minimum spanning Trees Example.
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Minimum spanning Trees Example..
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Minimum spanning Trees Example
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Minimum spanning Trees Example.
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Minimum spanning Trees Example..
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Minimum spanning Trees Example...
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Minimum spanning Trees Example.
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Minimum spanning Trees not unique
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Proof of correctness, Prims algorithm
Proof by induction At the i-th step we have
a partial MST outside the bag we select Least
weight edge crossing the bag
Light-edge
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Proof of correctness, Prims algorithm..
Assume Light-edge is not part of MST
gt Some other bag-crossing-edge must be part
of MST WHY?
gt ltp, light-edgegt cycle
gt cut heavy-edge, join light-edge ? reduce cost
(contradiction!)
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Concluding remarks
Minimum spanning Trees provide good starting
solutions For problems of the type connect
towns with roads, connect factories with supply
lines connect buildings with networks connect
town-areas with water/sewage channels For
real solutions extra (redundant) links may be
useful
next topic Transportation Planning Shortest
Paths